Stage 1 Physics B - Complete Waves Notes PDF

**5.1: WAVE MODEL** - Waves are periodic oscillations that transfer energy from one point to another. - In longitudinal waves, the direction of oscillation is parallel to the direction of travel of the wave. - In transverse waves, the direction of oscillation is perpendicular to...

**5.1: WAVE MODEL** - Waves are periodic oscillations that transfer energy from one point to another. - In longitudinal waves, the direction of oscillation is parallel to the direction of travel of the wave. - In transverse waves, the direction of oscillation is perpendicular to the direction of travel of the wave. - Represent transverse waves graphically and analyse the graphs. - Describe waves in terms of measurable quantities, including amplitude, wavelength (λ), frequency (*f*), period (*T*), and velocity (*v*). - Solve problems using: - [\$f = \\frac{1}{T}\$]{.math.inline} - [*v* = *fλ*]{.math.inline} Waves transfer energy from one place to another, but they do not transfer matter - For example, when light waves pass from a phone screen to your eye, only energy is being transferred. - For example: - Ripples transfer kinetic energy. - Sound waves transfer sound energy - To transfer from one place to another the waves vibrate (oscillate) - Oscillations -- a back-and-forth motion of an object between two points of deformation - For example, the oscillations of transverse waves is a up and down motion of an object. Waves can be divided into three categories: Mechanical wave: (Move through matter) - Motion requires a material medium which can be divided into three different types. - Each type *disturbs the medium* in a different way. Electromagnetic wave: (Move through a vacuum) - Do not require a medium in which to travel - They all travel through space at the speed of light. Particles vibrate back and forth parallel to the direction of energy transfer. - Oscillations that are parallel to the direction of energy transfer. - This leads to some regions that are more spread out and some that are more compressed because the wave is vibrating back and forth. - Note (for graphing): the oscillations move both left and right -- use arrows when labelling, as the direction of energy transfer will be horizontal. - Longitudinal waves move through matter/medium e.g. air, a liquid, or a solid. - As the energy moves it creates parallel motion - Once the energy passes, the particle returns to its rest position. Examples: - Sound waves travelling through air -- they travel as particles in the air move from side to side. - Sound waves are longitudinal and consist of a series of alternating compressions (the particles are closer together than in their normal/undisturbed state), and rarefactions (the particles are spaced further apart than normal). - If a red dot is on a slinky: (the red dot represent an air particle), the dot oscillates for side to side but it does not travel through the medium The particles vibrate about a mean (average) position perpendicular (at right angles) to the direction of energy transfer. - As the energy moves the particles vibrate perpendicular to the motion. Once the energy passes the particle returns to the rest position. - Transverse waves consist of a series of crests and troughs. Crests are often referred to as having a positive displacement and troughs a negative displacement. - On a graph the vibrations go up and down, while the wave is moving left to right. Examples: - Electromagnetic waves -- light and radio - Ripples and waves in water - Waves of string -- like in a guitar - If a red dot is on a slinky: when the wave travels the dot oscillates up and down but is does not travel along the medium. Note: not all transverse waves require a medium to travel. Transverse: A diagram of a curve Description automatically generated Longitudinal: For periodic waves there are certain characteristics common to all types: - Amplitude (*a* or *A*): the maximum displacement of a point from its equilibrium position - Movement from rest position. - Wavelength (λ): the distance between any two corresponding successive points; this could be the distance between two successive crest or two troughs (or between two successive compressions or two rarefactions) - Length of a single wave - Frequency (*f* or *n)*: the number of waves to pass a given point per second. It is also the number of complete vibrations per second of a point on the wave. Its units is hertz (Hz). - How many λ, or waves fit in 1 second. - One "up-down" - Period (T): the time for one complete wave to pass any point. It is also the time for one complete oscillation of a point on the wave. - Time for one λ to pass -- 1 cycle. - The time needed for the motion to repeat itself, that is, the time taken for one wave to pass a given point. A diagram of a graph Description automatically generated Frequency and period of a wave are related by the equation: \ [\$\$f = \\frac{1}{T}\$\$]{.math.display}\ Where, *f* is the frequency (measured in hertz, Hz), and T is the period (measured is seconds, s). - Wave velocity (*v*): the velocity of waves is related to other wave characteristics that represent the distance and time - Speed of the wave. - The distance (x axis) is how far the wave has travelled from its starting point, it is measured in metres. - The horizontal distance of one oscillation is called the wavelength. - This could be from the equilibrium -- up down and back up. - OR from the top of the wave (crest), down and back up to the next crest. - The displacement (y axis) is how far from the equilibrium point the wave has oscillated. - The maximum displacement is known as the amplitude (x axis to crest) - The graph may instead be a displacement time graph, therefore instead of measuring wavelength you are measuring time period -- this is the time it takes for one complete oscillation. - If you know the time period you can use the equation (f=1/T) to calculate the frequency (measured in Hertz, Hz or s-1), a measure of the number of complete oscillations per second. In the special case that the time interval is exactly one period, the wave would move a distance of one wavelength. - The velocity can be calculated from the equation: \ [\$\$v = \\frac{\\lambda}{T}\$\$]{.math.display}\ - This is more conveniently written as: \ [*v* = *fλ*]{.math.display}\ Where v is the velocity (m s^-1^), f is the frequency (Hz, or s^-1^), and λ is the wavelength (m). Perfectly Out of Phase: If one wave is shifted by half a wavelength (relative to the other), the troughs of one wave are aligned with peaks of the other and the waves are called perfectly out of phase - 90° phase shift - To calculate: λ/2.  1. How does the speed of a wave relate to its frequency and wavelength? The speed (v) of a wave it related to its frequency by the formula: [*v* = *fλ*]{.math.inline}. It is directly proportional to both its frequency and wavelength, meaning that if either increase, the speed also increases. 2. If you triple the frequency of a vibrating object, what happens to its period? If the frequency is tripled, the period decreases to one-third of its original value. This is because frequency and period have an inverse relationship: if the frequency increases, the period decreases. This is explained by the equation: f=1/T 3. How far, in terms of wavelength does a wave travel in one period? In one period, a wave travels a distance equal to its wavelength - To find the total time for a specific number of vibrations find T (number of vibrations per second) and multiply it by the number of vibrations given. - If you are told crests of waves are ()m apart, this is the wavelength. - s^-1^ is equivalent to Hz as the unit of frequency is: 1Hz = 1 cycle/second. - The period of vibration is T. - The largest possible value of the wavelength: subtract the two distances given (final -- initial). **5.2: MECHANICAL WAVES** - Mechanical waves, such as sound and seismic waves, transfer energy through a physical medium. - The natural frequency is the rate at which an object vibrates when it is disturbed by an outside force. - A forced vibration occurs when a wave forces an object to vibrate at the same frequency as the wave. - Resonance is the large-amplitude vibration that occurs in the object when the forced vibration is the same as its natural frequency. - Explain a range of wave-related phenomena including echoes, refraction, and resonance, using the mechanical wave model. - Use the principle of superposition of waves to explain a range of interference phenomena, including standing waves and beats. Mechanical waves are waves that require a medium to transfer energy from their source. - The particles in the medium are free to move a little from their position. They do not move forward together in bulk. - Mechanical waves can be transverse or longitudinal, however always travel in a material medium. - These material mediums are made of interconnected particles that are progressively disturbed. Note: in mechanical waves, energy is transferred through the medium. Examples: - Water waves, appear to be transverse waves (particles oscillate perpendicular to the direction of motion). - Sound waves, longitudinal waves (particles oscillate about mean positions in the same direction as the wave movement). Sound waves are longitudinal waves that travel through a medium, such as air, water, or solids, by means of a series of compressions and rarefactions. - Sound is produced by a vibrating source which causes the particles of the medium to vibrate in the direction in which the sound travels. - Sound is a compression wave that can be heard. Sound can be described in the same manner as other waves: - The frequency of sound, measured in Hz is called pitch. - The higher the pitch, the higher the frequency. A diagram of a function Description automatically generated with medium confidence - The speed that sound travels at is much slower than the speed of light. - Sound travels through air (at 20°C) with a speed of approximately 340m s^-1^. - The speed of sound differs when it is travelling through different substances. - Dependent on the material, and the temp. - i.e. fast in liquids, even faster in solids. - The denser the material sound energy is travelling in, the fast it moves. This is because vibrations are passed on more quickly. - Speed of sound in air increases by 0.6m s^-1^ for every degree rise in temperature. Seismic waves are mechanical waves -- vibrations that travel through the earth -- caused by the sudden movement of materials within the Earth, such as explosions and earthquakes. Two main shockwaves are associated with earthquakes: - P waves (Primary waves) -- fast moving longitudinal waves. As they reach Earth's surface (travel through Earth) the particles in the rocks are stretched apart and compressed in the direction of the wave, causing the ground to shudder. - Travel through liquids and solids - S waves (Secondary waves) -- slower moving transverse waves. Waves of energy that travel through Earth by causing particles in rocks to move at right angles to the direction of the wave, causing the rocks to vibrate from side to side. - Travel only through solids. P and S waves travel at different speeds, thus, there is a time delay between their arrival at the surface. - The P wave arrives first, then the S wave arrives. Surface wave arrives last -- to measure the time interval go from beginning of each wave's arrival. - This delay provides information about how far away the source is. - As P and S waves travel through the Earth, they may encounter regions and rocks of varying densities, causing the waves to change direction suddenly. - This is because of refraction, a phenomenon that occurs when waves pass from one medium to another with different physical properties, such as density. These changes alter the wave's speed, causing the wave to bend. - Earth is composed of layers (e.g. crust, mantal, outer core, inner core), each with varying densities and elastic properties. Seismic waves (both P, and S) encounter these boundaries and experience a change in speed. Thus, when they move from a less dense to a denser layer (or vice versa), their speed increases or decreases. - A faster wave bends away from the normal, while a slower wave bends toward the normal. - When two seismic wave pulses meet (e.g. during an earthquake where multiple wavefronts are generated), their displacement (amplitude) combine momentarily according to the principle of superposition (adding together if in the same direction or partially cancelling if in opposite direction). Afterward, the pulses continue traveling in their original directions, unchanged by the interaction. - If the displacements are in the same direction (constructive interference), the resultant displacement is larger. - If the displacements are in opposite directions (destructive interference), they partially or completely cancel each other out. Real world implications: - Two compressional P-waves meeting could amplify the ground's motion briefly (constructive interference). - A P-wave and an S-wave with opposing displacements could reduce the ground's motion temporarily (destructive interference). Free (natural) vibrations occur when an object is displaced from its equilibrium position and then left to vibrate by itself. Natural frequency is defined as the rate at which an object will vibrate when it is 'excited'/disturbed by an external force. For example: - A tuning fork: - Strike the fork, the prongs then vibrate about the mean position. - Elastic restoring forces strongly pull the prongs back and forth. - The tuning fork vibrates at its natural frequency. - Guitar strings - Organ pipes - Drums - Pendulums - Bungee jumping - A structure oscillates at a frequency that is not determined by its natural properties, but rather an external force. If a vibrating tuning fork is placed on a rubber stopper, it emits a low-intensity sound that can be heard only with difficulty. However, if the same vibrating tuning fork is held with its shaft on a wooden bench or tabletop, the sound is heard throughout the room. Because: - The fork causes the bench to vibrate with the same frequency. - The benchtop has a larger vibrating area than the tuning fork. - These forced vibrations disturb a greater volume of air and produce a louder sound. Forced vibrations doesn't occur with the rubber stopper because the rubber does not efficiently transmit the energy from the vibrating tuning fork to the stopper because: - For forced vibration to occur, the vibrating onject (in this case, the tuning fork), needs to transfer its vibrational energy to the object it's placed on (the rubber stopper). - The rubber stopper, being made of soft, flexible material, tends to absorb some of the vibrational energy rather than allowing it to pass through efficiently. - The rubber reduces the amplitude of the vibrations and doesn't transmit them effectively, limiting the vibrations in the stopper, and preventing it from vibrating strongly. - Even if some energy is transferred from the rubber stopper, the stopper itself has a small vibrating surface area. This means that, even if it does vibrate, the volume of air it can disturb is very small, and the sound produced is weak and difficult to hear. To determine whether forced vibration will occur with an object, we consider several key factors: - Material Properties (Transmission of Vibrations): - Hard, rigid materials (like wood, metal, or glass tend to transmit vibrations more efficiently. The vibrations pass through them with minimal energy loss, making forced vibrations more likely to occur. - Rigidity of the Object: - The object needs to be rigid enough to vibrate when forced. Rigid objects are more likely to vibrate in response to an external force because they can maintain their shape and vibrational energy for longer periods. - Surface Area and Contact: - If an object has a large contact area with the vibrating source (like a tuning fork on a wooden table), it can receive more energy and potentially vibrate more, producing a louder sound. Resonance occurs when an object is subject to a forced vibration (external force) that matches its natural frequency. When this happens, the object absorbs energy from the external force and stars vibrating with a larger amplitude. In more detail, if the driving force (the frequency at which an external force is applied to an object) is equal to the natural frequency of an object), then resonance occurs. Resonance is where the amplitude of oscillations of an object drastically increases due to gaining an increased amount of energy from the driving force. - When the natural frequency, and force of vibration match. Example -- What happens when you blow across the mouth of a bottle: - Air in the bottle vibrates and you hear a note. - The frequency of this note is determined by the dimension of the bottle. - The sound is a result of the free vibrations of the air in the bottle. By blowing air across the top of the bottle, provide waves of a number of frequencies to the air column inside the bottle. Most of the waves transfer energy inefficiently...however... Waves of one particular frequency, the natural frequency, transfer energy very efficiently and set up a standing wave in the bottle. The frequency of the standing wave is the frequency of the note you hear. Example -- singer shattering glass: - The key in this scenario is that the glass has a particular natural frequency, and the person is singing a note which is a forced frequency that is exactly the same. - The singer forces a vibration on the glass and the sound causes the glass to vibrate at its natural frequency, but because the glass is vibrating at its natural frequency and receiving energy from the forced vibration it is going to cause the glass to vibrate with a greater amplitude, causing the sound to be louder, thus breaking the glass. Sound waves do not just stop when it reaches, they end of the medium, or when it comes across an obstacle. Instead, the wave will undergo certain behaviours: - Reflection off the obstacle - Diffraction around the obstacle - Transmission (accompanied by refraction) into the obstacle or new medium REFLECTION: The law of reflection states that the angle of incidence is equal to the angle of refraction. ECHOES: REFRACTION: The change in direction of waves at the boundary between two different mediums. - Waves change direction when they strike a surface at an angle other than 90°. - The speed and wavelength of the wave also change. The frequency stays constant. Bending of the path of the waves, is a result of the change in speed and wavelength of the waves. If the medium (or its properties) are changed, the speed of the wave is changed. Waves passing from one medium to another will undergo refraction. For example: - Sound waves are known to refract when travelling over water. Even though the sound waves are not exactly changing medium, it is travelling through a medium with varying properties. Because water can affect the temperature of the air, and the speed of sound is slower in cooler air, sound is refracted.