Waves PDF

Waves By Dr. Elmughera Hussein Salim Elhag  The study of periodic vibrations and waves is one of the oldest scientific studies. Studies of wave motion also advanced through research into the behavior of light. Pulses on a Rope The simplest way to begin the study of waves is...

Waves By Dr. Elmughera Hussein Salim Elhag  The study of periodic vibrations and waves is one of the oldest scientific studies. Studies of wave motion also advanced through research into the behavior of light. Pulses on a Rope The simplest way to begin the study of waves is to think of the propagation of a pulse along a rope. If one end of a rope is snapped up and down to its initial position, the action generates a wave pulse that travels along the rope to the other end Definition A disturbance that transfers energy from one point to another without imparting net motion to the medium through which it propagates Wave types Waves are classified into two types: Transverse Waves: The disturbance is at right angles (perpendicular) to the direction of propagation of the wave. Pulses in a rope are an example of this type. Longitudinal Waves: The disturbance is parallel to the direction of propagation of the wave. Sound waves are example of this type.  Waves, also, can be classified into mechanical waves and electromagnetic waves. Mechanical waves need medium to transfer through, example of this type include sound waves, water waves, and pulses on a rope. Electromagnetic waves, on the other hand, do not need medium to transfer through, example of this type include light waves and radio waves. Representing Waves Graphically:  Wave Crest (Maxima): It is the maximum positive point from the equilibrium axis. Wave Trough (Minima): It is the maximum negative point from the equilibrium axis. Amplitude: It is the maximum displacement from the equilibrium axis. It is denoted by A. Its unit depends on the nature of the wave, In the case of pulses on a rope it is measured in meters, centimeters etc  Wave Length: It is the distance between two successive crests (or tow successive troughs). It is denoted by λ. It is measured in unit of length such as meters, centimeters, nanometers and so on. Periodic Time: It is the time for one complete oscillation. It is denoted by T. Its unit is second. Frequency: It is the number of oscillation per one second. Sometimes it is referred as linear frequency. It is denoted by f. Mathematically we can write: f =1 T  The Angular Frequency: When an object rotates in a circular motion it has an angular frequency. It is denoted by ω. The linear frequency, f, can be converted to its corresponding angular frequency by the following equation: ω= 2πf (2) The speed of a wave: The speed, v, of a wave is given by: v= λ.f (3) It is measured in meter/second, centimeter/second and so on.  The Phase Angle: When a wave is not passing through the origin (0,0) then it has a phase angle. It is denoted by φ. It is measured in radians. The phase angle is useful when we are handling more than one wave. The General Equation of Waves y= Asin(2π.x ± ωt+φ) (4) λ if we put: k=2π (5) , then we can write λ y= Asin(kx± ωt+φ) (6)  k is called the wave number and has the unit of (radians/meter, radians/centimeter, …) When the amplitude (A) of a wave is constant, the wave is called harmonic wave. Harmonic waves repeat their motion at constant intervals of time. Equation (6) represents the general equation of harmonic waves  When the phase angle (φ) of a wave is zero, equation (6) is written as: y= Asin(kx± ωt) (7) Equation (7) represents a harmonic wave that passes through the origin (0,0) like the wave in (Fig. 1) Example A harmonic wave traveling on a rope is represented by the equation: y=20sin(6x-30t+3.2) Taking the amplitude in meters and the wave length in centimeters find: the amplitude (A), the phase angle (φ), the frequency (f), the periodic time (T), the speed of the wave (v), and the direction of traveling. Solution Comparing the given wave equation with equation (6), we find: The amplitude, A=20 meters. The phase angle, φ= 3.2 radians. ω= 30 and since ω=2πf then f=30/2π Hertz. Periodic time T=1/f, then T=2π/30 seconds. To find the speed, we must, first, find λ: k=2π/λ, then λ=2π/k=2π/6 centimeters v= λ.f =2π/6×30/2π=5 centimeter/second Energy and Information Transfer by Waves: We experience the energy transferred by waves in many situations: our skin is warmed by the light waves from the sun, we hear sound waves, we feel the force of an ocean wave. Furthermore most of the information that we receive comes to us by waves. Speech and music are transmitted by sound waves  Radio and television are transmitted by electromagnetic waves. The reflected light by which you read this page is a wave. Generally, the information transmitted by waves is constructed from the wave parameters such as: the frequency (f), the amplitude (A), the wave length (λ), and the phase angle (φ). The energy (E) transmitted by a wave is directly proportional to the square of the amplitude (A), that is: E α A2  The energy is measured in joules. The power (P) is the rate of transmitting energy: P = ∆ E α A2 (9) ∆t ∆t The power is measured in watts The more generally useful parameter is the intensity (I), which is defined as “The power flowing through the unit area”, that is: Intensity (I) = Power (10) Area The intensity is measured in watt/meter2 Sound Waves Sound waves in air are longitudinal waves The speed with which sound waves propagate in air depends on atmospheric pressure, temperature, and humidity The speed of sound in dry air is 331.5 meter/second. It is much slower than the speed of light. The response of human ears is limited to a range of frequencies from 20 Hertz to about 20,000 Hertz. Those frequencies (20 Hz to 20,000 Hz) are referred to as audio, or sonic frequencies. In general, as we grow older the upper limit of audible frequencies drops.  Frequencies above 20,000 Hertz are beyond our hearing and are referred to as ultrasonic frequencies. Extremely low frequencies are called infrasonic frequencies. Ultrasonic vibrations generate sound like wave, but we do not hear them because of the limitations of our ears.  There are many applications of ultrasonic waves. Because of their short wave lengths, they can be focused into narrow beams and directed more easily than sound waves Ultra sonic imaging is used by physicians to look inside the human body. The visual image is reconstructed from the information contained in the amplitude and the phase of the reflected waves (See Fig.3) Ultrasonic imaging is generally considered a safer technique than the X-ray imaging. Intensity Levels of Sound Waves The human ear is extremely sensitive detector, capable of hearing sounds over an extremely large range of intensities. The minimum intensity that can be heard by human ears is 10-12 Watt/meter2.This intensity is called the threshold of hearing  Because the intensity of sound wave in Watts/meter2 is extremely small, a new scale is introduced for measuring the intensity of a sound wave. This scale is called the Intensity Level (IL). It is a logarithmic scale, its unit is called decibel (dB). This scale is given by: IL = 10×log(I/10-12) Where: IL≡ Intensity Level in decibel (dB) I≡ Intensity in Watts/meter2 Example Calculate the intensity level in decibels of a sound wave that has an intensity of 10-10 Watts/meter2 Solution IL = 10×log(I/10-12) IL = 10×log(10-10/10-12)= 10×2=20 dB A comparison of sound intensity levels due to various sources.  Listening to loud sounds over a period of time causes a temporary shift of the hearing threshold to higher intensity levels. The effect is a temporary hearing loss known as hearing fatigue. The duration and magnitude of the fatigue depend on both the intensity of sound causing it and the length of exposure time. If the exposure time and intensity are great enough, the hearing loss may become permanent instead of temporary.