Physics Transverse Waves PDF

SH1685 Mechanical Waves and Acoustics 2. A longitudinal wave is a wave whose vibrations travel along the Overview direction of th...

SH1685 Mechanical Waves and Acoustics 2. A longitudinal wave is a wave whose vibrations travel along the Overview direction of the wave. Thus, if a wave is moving to the right, the Harmonic motion, from its root word harmony, defines a kind of motion vibrations also move to the right. This wave creates areas of that shows repetition and definite pattern. This pattern is a wave, which compression where loose molecules clump together during is the driving mechanism of our songs, our sounds, and any other transmission at certain intervals. This wave travels fastest in a solid, sounds we create and associate with. With this fascination with sound which becomes nonexistent in a vacuum, where no molecules can comes acoustics. support the wave. An example of this wave is sound. Acoustics is the branch of science dealing with the study, analysis, and 3. The transverse wave is a wave whose vibrations travel application of sound, which is a disturbance of molecules within a given perpendicularly from the wave’s direction. It travels at certain substance. This disturbance must be caused by a wave, however. In this intervals and is usually drawn like a sine curve. It travels the fastest context, a wave is not just a graphical representation of harmonic in a vacuum, which becomes nonexistent in a solid, where the matter motion; rather, it is a medium that drives a substance to disturb its only absorbs the energy carried by the wave. An example of this molecules by use of harmonic motion. This harmonic motion is called wave is light, which is made up of two (2) perpendicular waves vibration. coexisting as one beam of light. Kinds of Waves Parts of a Transverse Wave A wave is a medium where sound propagates without carrying over the The crest is the highest point of the transverse wave, while its lowest matter where it moves. Waves do not only carry sounds, however. point is called the trough (pronounced as trof). Together, both the Different waves propagate different things, such as electrical signals, crest and trough form the wave’s amplitude. light, and so on. Such a wave is a mechanical wave. A wave’s wavelength is the distance between two (2) crests or A mechanical wave is a wave that results from interaction with matter. troughs. Its symbol is the Greek letter lambda (𝜆). An example would be a child driving a rope tied to a post to form waves. In this example, the child is delivering force onto the rope, which creates A wave’s frequency is the number of waves per measured time. a wave carrying the force. This force has a certain amount of energy Frequency is measured by counting the number of complete cycles traveling from the boy’s hand up to the tree. Thus, the wave dies if it hits of a wave within a second, with a complete cycle comprising of a fall the tree, and if the child has had enough of creating waves. This wave and rise of the wave. has a limited transmission of matter. Let us take a close look at these waves: 1. A surface wave is a wave that propagates along with the interface between different media. This mechanical wave creates a disturbance on the matter’s surface, hence the name. The wave generated carries over the energy of the source of the disturbance. As an example, if pressure builds up underneath the surface of the Earth, the resulting pent-up pressure generates surface waves along the planet’s surface, carrying over the energy stored. 10 Handout 1 *Property of STI  [email protected] Page 1 of 22  SH1685 A longitudinal wave demonstrating 1/8 period Parts of a Wave Speed (T), generating one (1) 𝑇. Longitudinal Wave Wave speed (𝑣) is the measure of how fast a wave propagates from the Two (2) given particles in source traveling outwards. It follows the same tenet as linear motion’s The given the medium, moving one (1) wavelength (𝜆) apart The compression is the velocity, where plunger densest point of the 𝑑𝑥 moves in from each other 𝑣=. longitudinal wave, while 𝑑𝑡 SHM its loosest point is called the rarefaction For wave speed, it depends on the derivative change of wavelength (pronounced as rair-uh within a given period, where + faction). The 𝑑𝜆 compression is the 𝑣=. 𝑑𝑇 longitudinal equivalent for the crest, and the Since period has the following derivation, rarefaction is the trough. As seen in 𝟏 𝑻= , Figure 1, darker areas 𝒇 show compression, where the fluid is at its it gives a new perspective on wave speed, which now becomes densest. 𝒗𝒘 = 𝝀𝒇 A wave’s wavelength is 𝝀 the distance between =. 𝑻 two (2) compressions or rarefactions. Its Mathematical Description of a Wave symbol is the Greek Waves can be described by the given concepts (i.e., wave speed, letter lambda (𝜆). amplitude, frequency, wavelength). Often, though, a more detailed description of the particles’ exact position in a medium of a propagating A wave’s frequency is wave is required. As an example, let’s set up a string with a very high the number of waves tension value. If we are to ignore the sagging of the string, then it is a per measured time. horizontal line. This line can now act as the abscissa of the system. The wave moves by a Frequency is measured Particles points oscillate with the single wavelength by counting the number Waves on a string are transverse, meaning waves traveling at the wave's amplitude during each 𝑇 of complete cycles of a abscissa has also displaced at the ordinate as well. The value of the (𝐴) wave within a second, ordinate now depends on the value of the wave on its abscissa. Figure 1. Periodic motion of a piston showing both with a complete cycle Mathematically, longitudinal and transverse wave in between two (2) comprising of a fall and 𝑦 = 𝑦(𝑥, 𝑡), points rise of the wave. Source: Sears and Zemansky's University Physics (with Modern Physics). 2016, p. 496 where the value of the ordinate (𝑦) is now dependent on the wave’s position at the abscissa (𝑥) to the wave’s period (𝑡). This now makes the 10 Handout 1 *Property of STI  [email protected] Page 2 of 22  SH1685 The given string has a set time interval of 1/8 period (𝑇), creating a total of one (1) 𝑇. ordinate a function of both 𝑥 “downstream” point) lags behind the movement of the particle on the left and 𝑡. The new function side by an amount equal to the distance between these particles. Wave Three (3) points given on a 𝑦(𝑥, 𝑡) is now called the generated by string, at least a half- wave function of the wave in This makes the cyclic motions of various points on the string out of step oscillator wavelength apart the system. With this, we with each other by various fractions of the cycle. Such differences are can now determine the called phase difference. A phase is a certain point in the system that displacement of any particle undergoes a certain event; thus, the phase of the motion is different for at any time (i.e., from different points. For example, if point A (see Figure 2) is at its highest equilibrium). From here, amplitude compared to point B at its lowest, then the two (2) are a half- velocity and acceleration cycle out of phase. can also be derived, as well as the string’s shape, and Mathematically, suppose that the displacement of a particle at the left many more. end of the string (𝑥 = 0), where the wave originates, is given by the equation As an example, suppose a 𝑦(0, 𝑡) = 𝐴 cos 𝜔𝑡 sinusoidal wave travels from = 𝐴 cos 2𝜋𝑓𝑡. left to right (where the value of 𝑥 increases) along a That is, the particle oscillates in simple harmonic motion with amplitude, string (see Figure 2). Every frequency, and angular frequency. The notation (𝑦(0, 𝑡)) reminds us that particle of the string the motion of this particle is a special case of the wave function that oscillates in simple describes the wave. At 𝑡 = 0, the particle at 𝑥 = 0 is at its maximum harmonic motion with the positive displacement (i.e., 𝑦 = 𝐴) and, instantaneously, is at rest due to same amplitude and 𝑦 being at maximum. frequency. But their oscillations are not in sync The wave disturbance travels from 𝑥 = 0 to some point in 𝑥 on the right with each other. As seen in side of the origin in an amount equal to 𝑥/𝑣, where 𝑣 is the wave speed. Figure 2, the particle at point So, the motion of point 𝑥 at a time 𝑡 is the same as the motion at point B is at its maximum value 𝑥 = 0 at the earlier time 𝑡 − 𝑥/𝑣. Thus, the displacement of 𝑥 and 𝑡 can (𝑦) at 𝑡 = 0 and returns to be determined by replacing 𝑡 by 𝑡 − 𝑥/𝑣. zero (0) at 𝑡 = 2⁄8 𝑇. These also occur to point A and to Mathematically, point C at 𝑡 = 4⁄8 𝑇 and 𝑡 = 𝑦(𝑥, 𝑡) = 𝐴 cos 𝜔𝑡 𝑥 6⁄ 𝑇, respectively, exactly = 𝐴 cos [𝜔 (𝑡 − )]. 8 𝑣 a half-period later. Thus, for any two (2) particles of a Since cos(−𝜃) is also equal to cos 𝜃, we can rewrite the equation as Figure 2. Tracking a three-point sinusoidal wave string, the motion of the generated via periodic motion particles on the right (in 𝑥 𝑦(𝑥, 𝑡) = 𝐴 cos [𝜔 (𝑡 − )] Source: Sears and Zemansky's University Physics terms of the wave, the 𝑣 (with Modern Physics). 2016, p. 497 10 Handout 1 *Property of STI  [email protected] Page 3 of 22  SH1685 𝑥 𝑥 𝑡 = 𝐴 cos [𝜔 ( − 𝑡)]. 𝑦(𝑥, 𝑡) = 𝐴 cos [2𝜋 ( − )] 𝑣 𝜆 𝑇 This displacement is a function of both location and time. The previous 𝑥 𝑡 = 𝐴 cos [2𝜋 ( − )] equation can still be generalized for both period (𝑇 = 1/𝑓) and 2𝜋 2𝜋 𝑘 𝜔 wavelength (𝜆 = 𝑣⁄𝑓 = 2𝜋𝑣⁄𝜔). Thus, 𝑘𝑥 𝜔𝑡 = 𝐴 cos [2𝜋 ( − )] 2𝜋 2𝜋 𝑥 2𝜋𝑘𝑥 2𝜋𝜔𝑡 𝑦(𝑥, 𝑡) = 𝐴 cos [𝜔 ( − 𝑡)] = 𝐴 cos [ − ] 𝑣 2𝜋 2𝜋 𝑥 𝑡 = 𝐴 cos(𝑘𝑥 − 𝜔𝑡). = 𝐴 cos [2𝜋 ( − )]. 𝜆 𝑇 A wave is also capable of traveling in the −𝑥 direction. In this case, the Now, let us introduce the wave number (𝑘), the spatial frequency of a displacement of a point 𝑥 at a given time 𝑡 is the same as the motion of wave. It is measured in cycles per unit distance (or in radians per unit point 𝑥 = 0 at a later time (𝑡 + 𝑥/𝑣). Thus, our wave equations become distance). Whereas frequency is the number of waves per unit time, the wave number is the number of waves per unit distance. Mathematically, 𝑥 𝑦(𝑥, 𝑡) = 𝐴 cos [𝜔 ( + 𝑡)] 𝑣 𝟐𝝅 𝑥 𝑡 𝒌= = 𝐴 cos [2𝜋 ( + )] 𝝀 𝜆 𝑇 = 𝐴 cos(𝑘𝑥 + 𝜔𝑡). or 𝟏 Thus, in the equation 𝑦(𝑥, 𝑡) = 𝐴 cos(𝑘𝑥 ± 𝜔𝑡), the quantity (𝑘𝑥 ± 𝜔𝑡) is 𝒗=. 𝝀 the wave’s phase. It plays a role as an angular quantity, and its value for any values of 𝑥 and 𝑡 determines what part of the sinusoidal cycle is Transposing the wave number equation to get the wavelength and occurring at any given point and time. As an example, if the value of 𝑦 = substituting it and the frequency equation into the wave speed equation, 𝐴 as a crest, then its phases can be 0, ±2𝜋, ±4𝜋, etc. provided that the we get cosine value is one (1). The values of 𝑦 = −𝐴, which is for the wave’s 𝑣 = 𝜆𝑓 trough, the phases then are ±𝜋, ±3𝜋, ±5𝜋, etc. 2𝜋 𝜔 = ( )( ) 𝑘 2𝜋 If the wave is shifted by 90°, we now have the equation, 𝜔 =. 𝑘 𝑦 = 𝐴 sin(𝑘𝑥 − 𝜔𝑡) 2𝜋 2𝜋𝑣 If the new wave speed equation were to be transposed to derive the = 𝐴 sin( 𝑥 − 𝑡) angular frequency, the equation is now for a periodic wave. 𝜆 𝜆 𝟐𝝅 Mathematically, = 𝑨 𝐬𝐢𝐧 (𝒙 − 𝒗𝒘 𝒕), 𝜔 = 𝑣𝑘. 𝝀 With the periodic wave derived, we can now rewrite the wave function equation into 10 Handout 1 *Property of STI  [email protected] Page 4 of 22  SH1685 because the shift of the wave is a derivation of the original one. Always But what happens if one (1) of the two (2) open ends is shut off? remember that for our lesson’s purpose, the variable 𝑥0 mentioned in the previous module is the same as 𝐴 in this module. This creates a close-end air column, a system that limits the frequency by forcing it to move to the other end, like how we create sound by Resonance blowing from an empty coke bottle. This creates a true half-standing Imagine riding a chain swing at the playground, with your body rocking wave pattern, resulting in the absence of even-numbered intervals. back and forth at a certain frequency (let us imagine that this setup Mathematically, experiences no damping). You are just there, swinging back and forth 𝟒 𝝀 = 𝑳. until a friend pushes you with the same velocity. From what you have 𝒏 guessed, your body begins to swing farther. That is because the swing experiences resonance. Fundamental Frequency A standing wave has a set amount of frequency in it. It can range from Resonance, in Physics, is a phenomenon that occurs when an object very low to very high. The lowest possible frequency for a standing wave with a natural frequency receives a forced vibration at a similar is known as its fundamental frequency, which can be determined by frequency. The effect of resonance varies -- the most common effect is the presence of nodes and antinodes. The nodes are the junctions in the shattering of glass due to the passage of sound waves. Different between frequencies, while antinodes are the waves themselves. As an objects have different resonance values. In this lesson, we can cover the example, if a string vibrates at the third harmonic, we get that there are following materials: strings, closed tubes, and open-ended tubes. three (3) antinodes and two (2) nodes. Mathematically, we get that, A string is any cord that is pulled taut, and its center is allowed to vibrate 𝒗𝒘 𝒇𝒏 = , when plucked, strummed, or any other action that will cause it to vibrate. 𝝀𝒏 When a string is plucked at a certain length, it creates a harmonic frequency that has a set number of crests and troughs. Mathematically, where 𝑛 = number of waves (or the antinode value). it can be represented as, 𝟐 The Wave Energy, Power, and Intensity 𝝀 = 𝑳, Every wave motion carries energy associated with it. The energy carried 𝒏 forth by both sunlight and earthquakes bears this out. To produce any where 𝑛 = the number of waves (or the wave’s interval value). wave, one has to apply force to a portion of the wave medium. The point where the force is applied moves, thus exerting work on the system, and An open-ended tube is a structure that allows frequencies to resonate therefore, has energy. As the wave propagates, each portion of the along the length of the pipe before going out, with flutes as an example. medium exerts a force and does work on the adjoining portions. In this When a frequency passes through one (1) of the open ends, it resonates way, a wave can transport energy from one (1) region of space to within the chamber, at half the wavelength. However, the frequency another. resonates twice within it, as if two halves of the wave are intertwined together. This creates a wave that behaves similarly to a vibrating string, Mathematically, the force has two (2) components (i.e. 𝐹𝑥 and 𝐹𝑦 ) that which is called an open-end air column. Mathematically, delivers the work on the system. Note that 𝐹𝑥 equates to the tension of the string, whereas 𝐹𝑦 /𝐹 equates to the negative of the string’s slope, 𝟐 and this slope is also given by 𝜕𝑦/𝜕𝑥. Putting these together, we get that 𝝀= 𝑳. 𝒏 10 Handout 1 *Property of STI  [email protected] Page 5 of 22  SH1685 𝜕[𝑦(𝑥, 𝑡)] we can also express the given equation in the alternative form of 𝐹𝑦 (𝑥, 𝑡) = −𝐹. 𝜕𝑥 𝑃(𝑥, 𝑡) = 𝐹𝑘𝜔𝐴2 sin2 (𝑘𝑥 − 𝜔𝑡) The value of 𝐹𝑦 at the point, 𝑦(𝑥, 𝑡) must be negative when the value of = √𝜇𝐹𝜔2 𝐴2 sin2 (𝑘𝑥 − 𝜔𝑡). the slope is positive. It is written as such to remind that its value may be different at different points along the string at different times. If the The sin2 function is never negative, so the instantaneous power in a specific point in the wave (which is assigned as point 𝑎) moves in the sinusoidal wave is either positive (so that the energy flows in the positive 𝑦 −direction, the force 𝐹𝑦 does work on 𝑎 and therefore transfers energy 𝑥 −direction) or zero (0, where the points have no energy transfer). into the part of the string to the right of 𝑎. The corresponding power (𝑃) Energy is never transferred in the direction opposite to the direction of at 𝑎 is the transverse force 𝐹𝑦 (𝑥, 𝑡) where 𝑎 multiplied with the transverse wave propagation. The maximum power of a wave occurs when the sin2 velocity 𝜐(𝑥, 𝑡) is of the function has the value, 𝑃(𝑥, 𝑡) = 𝐹𝑦 (𝑥, 𝑡)𝜐(𝑥, 𝑡) 𝜕[𝑦(𝑥, 𝑡)] 𝜕[𝑦(𝑥, 𝑡)] 𝑃(𝑥, 𝑡) = √𝜇𝐹𝜔2 𝐴2 , = −𝐹. 𝜕𝑥 𝜕𝑡 while the average value of sin2 function, averaged over any whole This power is the instantaneous rate at which energy is transferred along number of cycles, is 1/2. Hence, the string at 𝑥 and at time 𝑡. Note that energy is only transferred at points where the string has a nonzero slope (i.e., 𝜕𝑦/𝜕𝑥 is nonzero) so that the 1 𝑃𝑎𝑣𝑒 = √𝜇𝐹𝜔2 𝐴2. tension has a transverse component, and where the string has a 2 nonzero transverse velocity (i.e., 𝜕𝑦/𝜕𝑡 is nonzero) so that the transverse force can do work. Since energy is contained in the wave, it can be said that the wave’s power is traveling alongside the contained energy. Thus, The given equation earlier is valid for any wave on a string, sinusoidal, or otherwise. For a sinusoidal wave with a wave function, we have 𝐸 𝑃= 𝑡 𝑦(𝑥, 𝑡) = 𝐴 cos(𝑘𝑥 − 𝜔𝑡) 𝟏 = 𝝆𝑨⊥ 𝒗𝝎𝟐 [𝑨(𝒓)]𝟐. 𝜕[𝑦(𝑥, 𝑡)] 𝟐 = −𝑘𝐴 sin(𝑘𝑥 − 𝜔𝑡) 𝜕𝑥 𝜕[𝑦(𝑥, 𝑡)] Since all quantities on the right-hand side of this equation are = 𝜔𝐴 sin(𝑘𝑥 − 𝜔𝑡) independent of time, the power radiated by a wave is constant in time. 𝜕𝑡 𝑃(𝑥, 𝑡) = 𝐹𝑘𝜔𝐴2 sin2 (𝑘𝑥 − 𝜔𝑡). Note that this equation applies to both transverse and longitudinal waves. The energy of a mass in an oscillating spring is proportional to By using the relationships the spring constant multiplied to the square of the system’s amplitude, 𝜔 = 𝑣𝑘 which mathematically is 𝟏 𝑬 = 𝒌𝑨𝟐. and 𝟐 𝐹 𝑣2 = , 𝜇 10 Handout 1 *Property of STI  [email protected] Page 6 of 22  SH1685 We see that the energy of a wave is proportional to the square of the 𝑃 𝐼= amplitude and the square of the frequency. If the wave moves through 𝐴⊥ an elastic medium, the mass 𝑚 of the medium can be expressed as 1 𝜌𝐴⊥ 𝑣𝜔2 [𝐴(𝑟)]2 = 2 𝑚 = 𝜌𝑉, 𝐴⊥ 𝟏 = 𝝆𝒗𝝎𝟐 [𝑨(𝒓)]𝟐. where 𝜌 is the material’s density, and 𝑉 is the material’s volume. This 𝟐 volume is the product of the cross-sectional area (𝐴⊥ ) and the distance that the wave travels in time 𝑡. This gives the equation As we can see from this result, the intensity depends on the square of the wave’s radial amplitude (𝐴(𝑟)) from the point of propagation. For a 𝑙 = 𝑣𝑡. spherical wave in three (3) dimensions, the radial amplitude is inversely proportional to the radius of the distribution, which mathematically is This energy is distributed equally on the wave, whose value is described using this equation, 1 𝐴(𝑟) ∝. 𝟏 𝑟 𝑬 = 𝒎𝝎𝟐 [𝑨(𝒓)]𝟐 , 𝟐 Thus, where the amplitude of the wave has a radial dependence (i.e., 𝐴(𝑟)). 1 𝐼∝. Thus, the energy equation in this case is 𝑟2 1 Therefore, for two (2) points that are at distances of 𝑟1 and 𝑟2 from the 𝐸= 𝑚𝜔2 [𝐴(𝑟)]2 point of propagation of a spherical wave, the intensities of 2 1 = (𝜌𝑉)𝜔2 [𝐴(𝑟)]2 𝐼1 = 𝐼(𝑟1 ) 2 1 = (𝜌𝐴⊥ 𝑙)𝜔2 [𝐴(𝑟)]2 and 2 𝟏 𝐼2 = 𝐼(𝑟2 ) = (𝝆𝑨⊥ 𝒗𝒕)𝝎𝟐 [𝑨(𝒓)]𝟐. 𝟐 measured at these two (2) locations are related via This energy is used to determine the power described earlier. If a wave has energy and power, then it also has intensity. A wave’s intensity is 𝑰𝟏 𝒓 𝟐 =. the power radiating per cross-sectional unit. Mathematically, 𝑰𝟐 𝒓 𝟏 𝑷 Wave intensity can also be measured by employing decibels. A decibel 𝑰= 𝑨⊥ is a relative unit of measurement that is a tenth of a bel, a measure of 𝑷 wave power, on a logarithmic scale. Mathematically, this can be =. presented as 𝟒𝝅𝒓𝟐 Substituting the given power, the equation now becomes 10 Handout 1 *Property of STI  [email protected] Page 7 of 22  SH1685 𝑰 𝒅𝑩 = 𝟏𝟎 𝐥𝐨𝐠 ( ). 𝐖 𝟏 × 𝟏𝟎−𝟏𝟐 𝐦𝟐 Wave Superposition Up until this point, we have been discussing waves that propagate continuously in the same direction, both linear and radial. But when a wave strikes the boundaries of its medium, all or a part of it is reflected. When you yell at a building wall or a cliff’s face a few distances away, the sound wave is reflected from the rigid surface back to you, creating an echo. When you flick the end of a rope whose far end is tied to a rigid support, a pulse travels the length of the rope and is reflected to you. In both cases, the initial and reflected waves overlap each other in the same region of the medium. This is called interference, as it also refers to what happens when two (2) or more waves pass through the same region simultaneously. If a rope is tied to a rigid structure, that rope has a fixed end, and the wave it reflects travels in the opposite direction. The original pulse is called the incident pulse (whereas the returning pulse is the reflected pulse; its displacement is the opposite of the original). If the rope is free Figure 5. Wave superposition, creating two (2) forms of interference to move on its end, that end is a free end, and the reflected pulse travels Source: https://www.askiitians.com/iit-jee-wave-motion/interference-of-waves/ in the same direction as the incident wave. This creates the boundary conditions of the system. Because they have the same amplitude, the total displacement at their meeting point is essentially zero (0) at all times. Then, it creates a The Pulse Behavior reflected wave of the two, this time traveling in the opposite direction The formation of a reflected pulse is similar to the overlap of two (2) compared to their incident directions. These two (2) waves created a pulses traveling in the opposite direction. These converging pulses destructive interference with one (1) another. create a superposition, where the pulses overlap and pass each other; the total displacement of the string is the algebraic sum of the When it comes to Figure 5a, the two (2) waves are now traveling towards displacements at that point in the individual pulses. As seen in Figure each other in the same direction. As they collided at their midpoint, they 5b, it shows two (2) pulses with the same shape, with one (1) inverted. created a larger wave, which dissipates into the reflected waves, which are still in the same direction as before. This interference is called constructive interference, where two (2) waves of the same amplitude create a wave whose amplitude is the algebraic sum of the two (2) waves combined. Mathematically, the superposition of the waves can be described using the equation 10 Handout 1 *Property of STI  [email protected] Page 8 of 22  SH1685 𝒚(𝒙, 𝒕) = 𝒚𝟏 (𝒙, 𝒕) + 𝒚𝟐 (𝒙, 𝒕), Standing waves are also capable of both constructive and destructive interferences; the only downside is that they cannot transfer energy. We where the wave functions’ additive property follows from the form of the can derive a wave function for standing waves using the same equation wave equation, which is linear. 𝑦(𝑥, 𝑡) = 𝑦1 (𝑥, 𝑡) + 𝑦2 (𝑥, 𝑡). The Sinusoidal Behavior Compared to a pulse, a sinusoidal wave behaves differently. It can have Here, the wave 𝑦1 (𝑥, 𝑡) represents the incident wave traveling to the left superposition, but its fixed ends dictate its behavior. As an example, let’s along the positive side of the abscissa, arriving at the origin, and being imagine a string with a fixed left end. As the right end is moving the reflected. Meanwhile, the wave 𝑦2 (𝑥, 𝑡) is the reflected wave traveling to string, the resulting motion travels to the left and is reflected to the right. the right from the origin. We noted that the reflected wave from a fixed As simple harmonic motion is being applied to the right, more incident end of a string is inverted, thus waves are being created. However, these waves appear to have subdivisions in their superpositions. Compared to the pulses, whose 𝑦1 (𝑥, 𝑡) = −𝐴 cos(𝑘𝑥 + 𝜔𝑡) amplitude, wave pattern, and propagation speed remain constant, the sinusoidal waves created in this manner creates a wave with a constant and position, but the amplitude changes. 𝑦2 (𝑥, 𝑡) = 𝐴 cos(𝑘𝑥 − 𝜔𝑡). This particular behavior has waves creating nodes and antinodes, as The change in sign corresponds to the shift in phase of 180° (or 𝜋 seen in Figure 6. The node is the section in which the string never radians). At the origin, where 𝑥 = 0, the motion from the reflected wave moves, whereas the antinode is the section where the string moves and is 𝐴 cos 𝜔𝑡 and the motion from the incident wave is −𝐴 cos 𝜔𝑡, which where the amplitude is at its greatest. Because the wave pattern doesn’t can be rewritten as 𝐴 cos(𝜔𝑡 + 𝜋). From this, the wave function for the appear to be moving in either direction along the string, this kind of wave standing wave is the sum of the individual wave functions, pattern is called a standing wave. 𝑦(𝑥, 𝑡) = 𝑦1 (𝑥, 𝑡) + 𝑦2 (𝑥, 𝑡) = 𝐴[− cos(𝑘𝑥 + 𝜔𝑡) + cos(𝑘𝑥 − 𝜔𝑡)]. We can then rewrite each of the cosine terms by using the identities for the cosine of the sum and difference of two (2) angles, cos(𝐴 ± 𝐵) = cos 𝐴 cos 𝐵 ∓ sin 𝐴 sin 𝐵. Applying these and combining the like terms, we obtain the wave function for the standing wave, which is 𝑦(𝑥, 𝑡) = 𝐴[− cos(𝑘𝑥 + 𝜔𝑡) + cos(𝑘𝑥 − 𝜔𝑡)] = (2𝐴 sin 𝑘𝑥) sin 𝜔𝑡 = (𝑨𝑺𝑾 𝐬𝐢𝐧 𝒌𝒙) 𝐬𝐢𝐧 𝝎𝒕. Figure 6. Nodes and antinodes of a standing wave Source: http://macao.communications.museum/eng/exhibition/secondfloor/MoreInfo/2_11_0_StandingWave.ht ml 10 Handout 1 *Property of STI  [email protected] Page 9 of 22  SH1685 This equation shows that the standing-wave amplitude (𝐴𝑆𝑊 ) is twice the amplitude of either of the incident waves (i.e. 𝐴𝑆𝑊 = 2𝐴). This equation where 𝑛 stands for any positive integer (e.g., 1, 2, 3,...). That is, if a has a function of 𝑥 and 𝑡, and the factor 𝐴𝑆𝑊 sin 𝑘𝑥 shows the curvature string with length 𝐿 is fixed at both ends, a standing wave can exist if its of the sine curve. However, since the waves are standing waves, only wavelength satisfies the given equation. Now, solving this equation for 𝜆 the amplitude changes. We can use this equation to determine the and labeling the possible values of 𝜆 and 𝜆𝑛 , where 𝜆𝑛 is the wavelength position of the nodes present in the standing wave; these are the points at a certain node number, we find for which sin 𝑘𝑥 = 0, so the displacement is always zero (0). This occurs when 𝑘𝑥 = 0, 𝜋, 2𝜋, 3𝜋, …, or, using 𝑘 = 2𝜋/𝜆, 𝟐𝑳 𝝀𝒏 =. 𝒏 𝜋 2𝜋 3𝜋 4𝜋 5𝜋 𝑥 = 0, , , , , ,… Waves can exist on the string if the wavelength is not equal to one (1) of 𝑘 𝑘 𝑘 𝑘 𝑘 𝜆 3𝜆 5𝜆 these values, but there can’t be a steady wave pattern with nodes and = 0, , 𝜆, , 2𝜆, , … antinodes, and the total wave can’t be a standing wave, as seen in 2 2 2 Figure 6 -- these represent 𝑛 = 1, 2, and 3. A node exists at 𝑥 = 0, as there should be since this end is the fixed end of the system. Corresponding to the series of possible standing-wave wavelengths, 𝜆𝑛 is a series of possible standing-wave frequencies (𝑓𝑛 ), each related to its The Normal Modes corresponding wavelength by In discussing the standing wave, we assumed that the length of the 𝑣 𝑓𝑛 =. string is negligible. Now, let us analyze this standing wave with a definite 𝜆𝑛 length 𝐿, with both ends fixed. Such strings are found in musical instruments, such as guitars, violins, The smallest frequency, 𝑓1 , corresponds to the largest wavelength (i.e., and pianos. When a guitar string is plucked, it creates a standing wave the 𝑛 = 1 case), 𝜆1 = 2𝐿, thus, on its string; this wave is reflected and re-reflected from the ends of the 𝑣 𝑓1 =. string, making a standing wave. This standing wave, as a result, creates 2𝐿 sound whose frequency is determined by the properties of the string. Such wave creates a wave that corresponds to the entire fluctuations of This is called the fundamental frequency. The other standing-wave the string and is thus called the standing wave’s normal mode. The frequencies are normal mode also describes the sinusoidal movement of all particles in 2𝑣 𝑓2 = the string vibrating at the same frequencies. 2𝐿 𝑣 = To understand a standing wave’s normal mode, we first note that the 𝐿 3𝑣 standing wave must have a node at both ends of the string. We saw in 𝑓3 = , the previous topic that adjacent nodes are one-half of a wavelength (i.e., 2𝐿 𝜆/2) apart, so the length of the string must be 𝜆/2, or 2(𝜆/2), or 3(𝜆/2), and so on. These are all integer multiples of 𝑓1 , such as 2𝑓1 , 3𝑓1 , 4𝑓1 𝐴nd or in general some integer number of half-wavelengths, so on. We can thus express all these frequencies as 𝝀 𝒗 𝑳=𝒏 , 𝒇𝒏 = 𝒏 𝟐 𝟐𝑳 10 Handout 1 *Property of STI  [email protected] Page 10 of 22  SH1685 = 𝒏𝒇𝟏. 𝟏 𝑭 𝒇𝟏 = √. These frequencies, called harmonics, create the harmonic series. 𝟐𝑳 𝝁 Musicians usually call the higher fundamental frequencies as overtones. The table below describes the first four (4) overtones. This is also the fundamental frequency of the sound wave created in the surrounding air by the vibrating string. The inverse dependence of the Frequency Number Overtone Level frequency to the string length is illustrated by the long strings of the bass 𝑓2 First section of the piano as compared to the shorter strings of the piano’s treble section. A stringed instrument’s pitch is made by pressing a string 𝑓3 Second against the fingerboard with the fingers to change the length of the vibrating portion of the string. And finally, this explains how pulling the 𝑓4 Third string changes the pitch and frequency of the string. 𝑓5 Fourth Sound Table 1. The higher frequencies classified as overtones Previously, we have been discussing that sound is produced by the For a string with fixed ends at 𝑥 = 0 and 𝑥 = 𝐿, the wave function 𝑦(𝑥, 𝑡) compression and expansion of the medium. We have also been of the 𝑛th standing wave is given by the equation discussing that it is also generated from the vibrations of an object, which generates sound. From our common knowledge, sound waves always travel outward in all directions from the source, with an amplitude that 𝑦(𝑥, 𝑡) = (𝐴𝑆𝑊 sin 𝑘𝑥) sin 𝜔𝑡, depends on the direction and distance from the source. From the study of the wave function, sound can also be idealized by allowing it to which satisfies the condition that there is a node at 𝑥 = 0. With 𝜔 = 𝜔𝑛 , propagate in the positive 𝑥 −direction only. As discussed earlier, for such which is also equal to 2𝜋𝑓𝑛 , and with 𝑘 = 𝑘𝑛 = 2𝜋/𝜆𝑛 , the new wave a wave, the function 𝑦(𝑥, 𝑡) gives the instantaneous displacement 𝑦 of a equation now becomes particle in the medium at position 𝑥 at a given time 𝑡. If the wave is sinusoidal, we can then express sound as 𝒚(𝒙, 𝒕) = (𝑨𝑺𝑾 𝐬𝐢𝐧 𝒌𝒏 𝒙) 𝐬𝐢𝐧 𝝎𝒏 𝒕. 𝑦(𝑥, 𝑡) = 𝐴 cos(𝑘𝑥 − 𝜔𝑡). From the fundamental frequency’s equation, a vibrating string’s frequency is equal to 𝑣 As also stated earlier, the sound is a longitudinal wave whose 𝑓1 =. displacement is parallel to the direction of travel of the wave, so the 2𝐿 distances 𝑥 and 𝑦 are measured parallel to each other, not The speed of waves on a string is determined by perpendicular. The amplitude 𝐴 here is the maximum displacement of a particle in the medium from its equilibrium position (see Figure 1). Hence, this amplitude is also called the displacement amplitude. 𝐹 𝑣=√. 𝜇 Sound is usually described as the pressure at various points. In a sinusoidal wave in the air, the pressure fluctuates sinusoidally above or Combining these equations yields us below the atmospheric pressure (𝑃𝑎 ) with the same frequency as the motions of the air particles. We sense these fluctuations by our eardrums 10 Handout 1 *Property of STI  [email protected] Page 11 of 22  SH1685 moving to the pressure difference from the external pressure generated The fractional volume change is related to the pressure fluctuation by by the sound and the body’s internal pressure. the bulk modulus 𝐵, which is 𝐵 = −𝑃(𝑥, 𝑡) > (𝑑𝑉⁄𝑉 ). Solving for 𝑃(𝑥, 𝑡), we have To better understand this, let 𝑃(𝑥, 𝑡) be the instantaneous pressure 𝜕𝑦(𝑥, 𝑡) fluctuation in the sound wave at any point 𝑥 at time 𝑡. That is, 𝑃(𝑥, 𝑡) is 𝑃(𝑥, 𝑡) = −𝐵 the amount by which the pressure differs from normal atmospheric 𝜕𝑥 = 𝐵𝑘𝐴 sin(𝑘𝑥 − 𝜔𝑡). pressure 𝑃𝑎. Think of 𝑃(𝑥, 𝑡) as a gauge pressure -- it can be either positive or negative. The absolute pressure at a point is then 𝑃𝑎 + 𝑃(𝑥, 𝑡). It shows how individual particles of the wave are displaced at this time. To see the connection between the pressure fluctuation 𝑃(𝑥, 𝑡) and the While 𝑦(𝑥, 𝑡) and 𝑃(𝑥, 𝑡) describe the same wave, these two functions displacement 𝑦(𝑥, 𝑡) in a sound wave propagating in the positive are one-quarter cycle out of phase: At any time, the displacement is 𝑥 −direction, consider an imaginary cylinder of a wave medium (gas, greatest where the pressure fluctuation is zero and vice versa. In liquid, or solid) with cross-sectional area 𝑆 and its axis along the direction particular, note that the compressions (points of greatest pressure and of propagation. When no sound wave is present, the cylinder has length density) and rarefactions (points of lowest pressure and density) are ∆𝑥 and volume 𝑉 = 𝑆∆𝑥, as shown by the shaded volume in Figure 1. points of zero displacements. The new equation shows that the quantity When a wave is present, at time 𝑡 the end of the cylinder that is initially 𝐵𝑘𝐴 represents the maximum pressure fluctuation. We call this the at 𝑥 is displaced by 𝑦1 = 𝑦(𝑥, 𝑡), and the end that is initially at 𝑥 + ∆𝑥 is pressure amplitude, denoted by 𝑃𝑚𝑎𝑥. Thus, displaced by 𝑦2 = 𝑦(𝑥 + ∆𝑥, 𝑡). If 𝑦2 > 𝑦1 , the cylinder’s volume has increased, which causes a decrease in pressure. If 𝑦2 < 𝑦1 The cylinder’s 𝑷𝒎𝒂𝒙 = 𝑩𝒌𝑨. volume has decreased, and the pressure has increased. If 𝑦2 = 𝑦1 , the cylinder is simply shifted to Waves of shorter wavelength 𝜆 (larger wave number 𝑘 = 2𝜋/𝜆) have the left or right; there is no volume change and no pressure fluctuation. greater pressure variations for a given displacement amplitude because The pressure fluctuation depends on the difference between the the maxima and minima are squeezed closer together. A medium with a displacements at neighboring points in the medium. Quantitatively, the large value of bulk modulus B is less compressible and so requires a change in volume 𝑉 of the cylinder is greater pressure amplitude for a given volume change (that is, a given displacement amplitude). ∆𝑉 = 𝑆(𝑦2 − 𝑦1 ) = 𝑆[𝑦(𝑥 + ∆𝑥, 𝑡) − 𝑦(𝑥, 𝑡)]. Since the bulk modulus is mentioned, it gave an idea that sound travels in a fluid, which in this case is water. Its speed is calculated using the In the limit ∆𝑥 → 0, the fractional change in volume 𝑑𝑉/𝑉 (volume equation change divided by original volume) is 𝑩 𝒗=√ , 𝑑𝑉 𝑆[𝑦(𝑥 + ∆𝑥, 𝑡) − 𝑦(𝑥, 𝑡)] 𝝆 = lim 𝑉 ∆𝑥→0 𝑆∆𝑥 𝜕𝑦(𝑥, 𝑡) =. where 𝜌 is the density of the fluid. If sound travels in a solid, the 𝜕𝑥 material’s Young's modulus is used. Thus, 𝒀 𝒗=√. 𝝆 10 Handout 1 *Property of STI  [email protected] Page 12 of 22  SH1685 If sound travels in a gas, the equation now becomes, at a standard temperature 𝑇, FLUID MECHANICS Overview 𝜸𝑹𝑻 𝒗=√ , We always experience fluids every single day. But, if we are to study 𝑴 how the term "fluid" is understood in the vernacular, we always mean it as a liquid. To Physics, however, fluid is any substance that where 𝛾 is the ratio of heat capacities of the gas, 𝑅 is the ideal gas characterizes flow, i.e., any substance that has no rigid structure is constant, 𝑇 is the absolute temperature (i.e., absolute means always amorphous and conforms to the shape of the receptacle it is in. positive in value), and 𝑀 as the gas's molar mass. The value of 𝑅, as a constant, is In its broadest sense, a fluid can either be a liquid or a gas. If we are to 𝑹 = 𝟖. 𝟑𝟏𝟒 𝐉/𝐦𝐨𝐥 ⋅ 𝐊. recall our basic science, a liquid has no shape but is denser than a solid, whereas a gas has no shape and is the least dense of the four (4) states If the sound travels in the air but is affected directly by temperature (except plasma, which is lighter and has more energy). This is because alone, then the equation becomes the atoms in both liquid and gas can freely move, as they have higher energy compared to a solid. 𝐦 𝑻 Fluid Properties 𝒗 = 𝟑𝟒𝟑 ∙√ , 𝐬 𝟐𝟕𝟑. 𝟏𝟓 𝐊 Density Density (𝜌, Greek letter “rho”) is defined as the amount of mass m per where 𝑇 is the temperature on the Kelvin scale. unit volume V of a substance. It can be computed by simply dividing the mass of the object by its volume. Mathematically, The Doppler Shift 𝒎 Sound, as a constant fluctuation of vibrations and pressure changes, can 𝝆= also experience changes in itself, especially if it's moving. This 𝑽 𝑷 phenomenon is known as the Doppler effect, where sound intensity = , increases or decreases depending on the wave speed of the moving 𝒉 source. If the sound source is moving towards you, then where 𝑃 is pressure, and ℎ is the thickness in meters. 𝒗 𝒇=[ ]𝒇 , The SI unit of density is kg/m3. This unit is awkwardly large for some 𝒗 − 𝒗𝟎 𝟎 solids or liquids. Therefore, other units of similar dimensions such as g/cm3 and g/ml are used more often. Density is an intensive property where 𝑓0 is the source frequency, 𝑓 is the final frequency heard by the and does not depend on the quantity of mass present, i.e., for a given observer, 𝑣 is the wave speed of the sound heard by the observer, and material, the ratio of mass to volume always remains the same. It can 𝑣0 is the original wave speed of the sound coming from the source. be concluded that volume 𝑉 is directly proportional to mass 𝑚. Likewise, if the source of the sound is moving away from you, then 𝒗 Density is not restricted to fluids -- solids also have densities since they 𝒇=[ ]𝒇. both have mass and volume. 𝒗 + 𝒗𝟎 𝟎 10 Handout 1 *Property of STI  [email protected] Page 13 of 22  SH1685 Specific Gravity kinematic viscosity (𝜂𝜈 ). Kinematic viscosity is the ratio of a fluid's A fluid's specific gravity (𝑔𝑠𝑝 ) is the ratio of a fluid's density compared to viscosity and its inherent density. The unit of kinematic viscosity is the a standard fluid density. Mathematically, Stoke (St), named after the Irish physicist George Stokes. One (1) stoke is equal to one (1) cm2/s, which in turn is 0.00001 m2/s. This makes one 𝝆 (1) m2/s equal to 10,000 stokes. 𝒈𝒔𝒑 =. 𝝆𝒔𝒕 The Case For liquids, the standard density (𝜌𝑠𝑡 ) used is water, whose density is Fluids, as free-flowing materials, are found in almost any Physics and/or 1 g/cm3. For gases, the standard density is air (i.e., a mixture of various other science-related cases. The only downside is that there is elemental and compound gases), whose density is 1.29 g/L. something that we need to know -- the fluid's flow. Fluids exist only in two (2) states -- static and dynamic. Pressure Pressure (P) is a measure of how much force is applied over a given The Static Case (Fluid at Rest) area. It is the magnitude of the force on a surface per unit area. The A fluid is considered to be static if there is no change in a fluid contained force is perpendicular to the surface area. It can be written as in a receptacle. This means that the density is constant and uniform throughout its volume. This is one (1) of the characteristics of an ideal 𝑭 fluid. Real fluids are, in truth, more complicated to deal with. 𝑷=. 𝑨 The density changes with varying temperatures. However, one can use This property is similar to another concept in our previous lesson. For the ideal liquid as an approximation to real liquids, granted that the other solids, this is called stress. The units of pressure are Pa or N/m2 (1 Pa = quantities that make the computation more complicated could be 1 N/m2 or 1kg /m s2), dyne/cm2, or any similar unit with the same ignored, such as when the temperature is constant or almost constant. dimension. Another frequently used unit of pressure is the atmosphere Ideal liquids have other characteristics -- but for a liquid at rest, the only (atm). assumption needed is that the density is constant. Thus, the pressure of a static fluid is computed to be at Viscosity Viscosity (𝜂, Greek letter "eta") is the measure of how it resists flowing. 𝑃 = 𝜌𝑔ℎ. Physically, it is the ratio of shearing stress (𝜏) to the velocity gradient in a fluid. Liquids such as water and cooking oil flow easily because of their As one can observe, the equation above is similar to the equation of low viscosity, while thick fluids such as honey and condensed milk tend gravitational potential energy, to resist flowing. The unit of viscosity is pascal-second (Pa ⋅ s), but it is 𝑈𝑔 = 𝑚𝑔ℎ. rarely used in the scientific community, preferring the Poise [pronounced pwahz] (P) and the dyne-s/ cm2 over it. The unit Poise is named in honor Only in this case, density (𝜌) acts as the mass. Also, take note of how of the French physiologist Jean Poiseuille. Ten (10) poise is equal to one the pressure, demonstrated as to how the property is exerted by a liquid (1) Pa ⋅ s, making the centipoise (cP) equal to a millipascal-second (mPa ⋅ at the base of its container, is related to its weight because weight is s). simply the force 𝐹 exerted on the base and area 𝐴 is the area of the base. There is another equation useful in similar problems, which will be Viscosity can be influenced by several applied forces. If only gravity is derived. Look at the equation of pressure, and note that the force is the the acting applied force to the viscous fluid, then it is experiencing a weight of the liquid 10 Handout 1 *Property of STI  [email protected] Page 14 of 22  SH1685 𝐹 = 𝑚𝑔. Since both sides of the equation contain the variable for mass, the variables are therefore canceled. Thus, we now get Substituting this in the pressure equation, we get 1 𝑔 𝐹 = 𝑃= 𝜌ℎ 𝑃 𝐴 𝑷 = 𝝆𝒈𝒉. 𝑚𝑔 =. 𝐴 With this equation, one can find the liquid pressure at any depth in the container, not just at the bottom. The only variable here is the height h Now, we introduce the density equation, of the liquid above a certain depth. The constant 𝑔 is simply the 𝑚 acceleration due to gravity, while 𝜌 is simply the density of the liquid. 𝜌=. This implies that liquid pressure does not depend on mass and volume, 𝑉 rather on height. We already know that volume is a 3D representation of the area with an added dimension -- thickness (ℎ). Given any 2D construct with a given Consider point A in the given diagram. Point A area (𝐴), adding thickness to it gives rise to volume. Thus, shows a particular location inside the partially filled container. The pressure at point A does not 𝑉 = 𝐴ℎ. depend on the actual mass or volume of the liquid surrounding it. It only depends on the height of the Substituting this in the density equation, we get,  h liquid above point A and the density of the liquid. A Hence, the locations deeper than point A 𝑚 experience more pressure, and those shallower 𝜌=. than point A experience less pressure. 𝐴ℎ Rearranging both new equations to get the area, we now get The Principles of Fluid Statics Pascal's Principle 𝑚𝑔 Blaise Pascal noted in one (1) of his studies that pressure applied to a 𝑃= fluid in a closed container is transmitted equally to every point of the fluid 𝐴 𝑚𝑔 and the walls of the container. This is essentially proven to be true and 𝐴=. 𝑃 is now called Pascal's principle. As for density, 𝑚 𝜌= 𝐴ℎ 𝑚 𝐴=. 𝜌ℎ Combining both equations, we now get 𝑚 𝑚𝑔 =. 𝜌ℎ 𝑃 10 Handout 1 *Property of STI  [email protected] Page 15 of 22  SH1685 Suppose there is a container that is partially filled with liquid up to a certain height A. Next, drop something into the liquid. This could be anything; a cork, a piece of rock, a gold nugget, etc. When that object is dropped on the A liquid, some of the liquid will rise. Since something was put into the liquid, expect some of the liquid to be displaced. In the illustration provided, some of the liquid rose above point A. Based on Archimedes’ principle, the force that is pushing that object up is equal to the weight of the liquid above point A. Figure 1. Pascal's Principle in action Source: http://pascalteam.hu/en_pascal_law.php In the illustration, the buoyant force (𝐹𝐵 ) is the magnitude F A of force pushing the object up, which is also the B Pascal’s principle can be easily demonstrated using the illustration magnitude of the weight of the liquid above point A. 𝐹𝑊 F above. In Figure 1, the liquid is confined. Hence, based on Pascal’s is the weight of the object. In most cases, the volume of W Principle, the pressure on the left piston should be equal to the pressure the liquid displaced is equal to the volume of the object submerged. The on the right piston. Mathematically, exception here is when the object has a cavity or when the object can absorb the liquid. 𝑷𝑳 = 𝑷𝑹. Archimedes’ principle is useful, especially in determining which object Since pressure is defined as force per unit area, we get will float and which object will sink. There are three (3) possible scenarios that could happen in case an object is submerged in a fluid: 𝑭 𝑭 The weight of the object is greater than the weight of the fluid ( ) =( ). 𝑨 𝑳 𝑨 𝑹 displaced (𝐹𝑊 > 𝐹𝐵 ). Hence, the object will sink. The weight of the object is equal to the weight of the liquid Archimedes’ Principle displaced (𝐹𝑊 = 𝐹𝐵 ). Hence, the object will float. Archimedes’ principle, named after the Greek natural philosopher The weight of the object is less than the weight of the object Archimedes, states that any object completely or partially submerged in displaced (𝐹𝑊 < 𝐹𝐵 ). fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid displaced by the object. Hence, the object will move upward until it is only partially submerged. When the object is only partly submerged, some of the displaced liquid molecules will go back to their original location, causing the buoyant force to decrease until it is equal to the weight of the object. The object will float while being partially submerged. Mathematically, we can describe buoyancy as 𝑭𝑩 = 𝑭𝟐 − 𝑭𝟏 = 𝝆𝒇𝒍 𝒈𝑽, where the force is computed as derived from the pressure equation, 10 Handout 1 *Property of STI  [email protected] Page 16 of 22  SH1685 𝐹 2. Laminar flow 𝑃= 𝐴 The flow of fluids is said to be steady or laminar if the velocity of 𝑭 = 𝑷𝑨 moving fluid at a fixed point does not change. For instance, = 𝝆𝒈𝒉𝑨, water molecule A passes through point O (represented as a small dashed circle) with a speed of one (1) m/s due east (see and 𝜌𝑓𝑙 = fluid density. Figure 1). Twenty seconds later, another molecule, B, passes through point O. The flow is steady if molecule B passes through Archimedes' principle also notes how the difference of the object's mass point O at a speed one (1) m/s due east (no change in (𝑚0 ) to its suspension mass (𝑚𝑠 ) is equal to the fluid's displaced mass. magnitude and no change in direction). This is computed by 𝒎𝟎 − 𝒎𝒔 = 𝝆𝑽, 3. Newtonian flow The viscosity of a fluid dictates the rate of flow of a fluid. If one where the product of both the fluid's density and volume equates to the dips their fingers in a Newtonian fluid, there is absolutely nothing fluid's displaced mass. to slow one down. John William Strutt, a British scientist who was also 3rd Baron Rayleigh, noted that a boat moving through The Dynamic Case a Newtonian fluid will never come to a stop even if all engines A fluid is considered to be dynamic if there is motion involved or if there were shut down. However, a boat at rest floating on a Newtonian are changes in the system. The motion of real fluids is very complicated fluid will not be able to move using its propellers also due to the and not yet fully understood. An example of a very complicated motion smooth flow of the fluid. is that of smoke that rises and changes its flow from steady to turbulent. As a counterpoint, some fluids, like water flowing through a hose, are a Figure 2. The demonstration of laminar flow close approximation of an ideal dynamic fluid. 4. Irrotational flow The following are the four (4) assumptions about ideal fluids: Another characteristic of an ideal dynamic fluid is that the flow 1. Incompressible flow is irrotational. To determine this property, allow a small particle The similar assumption to fluids at rest; the density is constant of dust to float on the fluid. The dust may or may not move in a and uniform. circular path, but it will certainly not rotate about an axis through its center of mass. If the particle rotates, then the flow is not A B A B A B irrotational. t =0s O O O The Continuity Equation Whenever watering plants in the garden or washing a car, one may have A B A B A B noticed that the speed of water flowing out of the hose can be increased t = 20 s by partially closing the hose’s end with one’s thumb. One can see that O O O the speed of the water is affected by the cross-sectional area of the tube, hose, or any means of its passage. An expression that relates the speed steady not steady not steady of ideal fluid to the cross-sectional area of the duct shall not be derived. same speed same speed same direction same direction different direction different speed 10 Handout 1 *Property of STI  [email protected] Page 17 of 22  SH1685 A region within a tube with varying cross-sectional areas will be studied. (𝑨𝒗)𝟏 = (𝑨𝒗)𝟐. The region is defined by length 𝑙. A certain volume of ideal fluid 𝛥𝑉 enters the region from the left within time 𝛥𝑡 (see Figure 2). Alternatively, it can be expressed as 𝑙 (𝑨𝒗)𝒊𝒏 = (𝑨𝒗)𝒐𝒖𝒕 𝑨𝒊𝒏 𝒗𝒊𝒏 = 𝑨𝒐𝒖𝒕 𝒗𝒐𝒖𝒕. 𝑡 𝐴1 𝒗𝟏 𝐴2 𝒗𝟐 Flux, when verbally expressed, implies that an ideal fluid flowing through a tube of varying cross-sectional areas has a constant volume flow rate. Another way of expressing flux is in terms of volume flow rate (ℱ) and mass flow rate (𝑅𝑚 ). Mathematically, 𝐴1 𝒗𝟏 𝐴2 𝒗𝟐 𝑡 + ∆𝑡 𝑨𝒗 = 𝓕 𝑹𝒎 =. 𝝆 Figure 3. The demonstration changes in fluid flow based on the pipe's cross-sectional area Energy In the given figure, 𝑣1 is the velocity of ideal fluid that is entering the Fluids also demonstrate energy. It is derived by dividing the volume into region while 𝑣2 is that of exiting the region; 𝐴1 is the cross-sectional area the equation. For a fluid's potential energy, it is derived by dividing of the left portion of the tube while 𝐴2 is that of the right portion. O