Waves Part 1: Travelling Waves Last modified: 15/05/2018
Links Contents Travelling Waves Harmonic Waves Wavelength Period & Frequency Summary Example 1 Example 2 Example 3 Example 4 Transverse & Longitudinal Waves Waves on a String Transverse Velocity Transverse Acceleration Wave Speed Power Transmitted Example Summary
Travelling Waves Contents A travelling (or progressive) wave is a disturbance in a medium that moves while maintaining its shape over time. This could be an ocean wave, a wave on a string, a sound wave etc. While the details of these differen types of wave vary, the basic properties are the same and will be discussed in this lecture. The wave disturbance can have any shape, but the condition that this shape doesn t change puts some limits on the mathematical description of the wave. A one-dimensional wave, moving in the x-direction can be described by a wave function y(x, t) which describes the magnitude of the disturbance at position x and time t. For a wave on a string y would be the distance away from the straight line of the string that the wave has moved the string at that time.
v x y(x, t) If a wave is to maintain its shape after moving a distance x in time t, then we must have: y(x, t) = y(x + x, t + t) = y(x + v t, t + t) x = v t v y(x + x, t + t) This can only be true if: y(x, t) = F (x vt) where F can be any function at all. (F will determine the exact shape of the wave.
We have assumed here that the wave is travelling in the +x-direction - as t increases, so does x. If instead the wave is travelling in the x-direction, we use similar logic to find: y(x, t) = y(x x, t + t) = y(x v t, t + t) which requires: y(x, t) = F (x + vt) where again F can be any function. A travelling wave in the +x direction is described by: y(x, t) = F (x vt) A travelling wave in the x direction is described by: y(x, t) = F (x + vt) where in each case, the speed v is a positive number.
If we have two waves travelling in the same medium at the same speed and direction: y 1 (x, t) = F 1 (x vt) and y 2 (x, y) = F 2 (x vt) Then the disturbances caused by the two waves can be added: y(x, t) = y 1 (x, t) + y 2 (x, t) = F 1 (x vt) + F 2 (x vt) = G(x vt) y is again a function of x vt, so is also a travelling wave, called the superposition of the two waves. The same principle will apply to three, four, five etc waves. A very useful technique is to do this procedure in reverse - expressing a wave as the superposition of harmonic waves, where F is the sin function. Because of this, we will focus on the properties of harmonic waves in the following.
Harmonic Waves Contents The general equation for a harmonic wave is: y(x, t) = A sin (k(x vt) + φ) where A, k and φ are constants. This is often written using ω = kv as: y(x, t) = A sin (kx ωt + φ) At t = 0 this looks like: y φ k A A x Since the maximum value of sin is 1, A represents the maximum value of the disturbance, or the amplitude of the wave. φ is the starting phase - it indicates the starting point of the wave. Often this is unimportant and can be assumed to be zero.
Wavelength Contents An important property of a wave is the wavelength, λ, which measures the distance between two consecutive peaks of the wave (or actually between any point, and the next repeat of that point). y λ y λ λ x x + λ x λ λ x Mathematically, this means: y(x, t) = y(x + λ, t) A sin(kx ωt + φ) = A sin(k(x + λ) ωt + φ) = A sin((kx ωt + φ) + kλ)
Clearly, we must have: kλ = 2π And so, the wavelength λ is connected to the wave number k by: kλ = 2π λ = 2π k The term wave number is used for historical reasons, don t stress about what it means. The SI unit of wavelength, λ is of course metres (m) The SI unit of wave number, k is metres 1 (m 1 )
Period & Frequency Contents Another important property of a wave is the period, T, which measures the time taken for a point on the string to complete a full up and down oscillation. t = 0 t = 3 8 T t = 3 4 T t = 1 8 T t = 1 2 T t = 7 8 T t = 1 4 T t = 5 8 T t = T
Mathematically: y(x, t) = y(x, t + T ) A sin(kx ωt + φ) = A sin(kx ω(t + T ) + φ) = A sin((kx ωt + φ) ωt ) ωt = 2π and the angular frequency, ω is connected to the period: where f = 1 T ω = 2π T = 2πf is the frequency of the wave. The SI unit of period, T is of course seconds (s) The SI unit of angular frequency ω is Hertz (Hz) (1 Hz = 1 s 1 ) Frequency f is also measured in Hertz.
Harmonic Waves Summary Contents The general equation for the displacement caused by a harmonic wave is: y(x, t) = A sin (±kx ± ωt + φ) where: A is the amplitude of the wave k is the wave number (always positive), the wavelength λ is calculated from k: λ = 2π k ω is the angular frequency (also always positive), the period T and frequency f are calculated from ω: T = 2π ω f = 1 T = ω 2π φ is the starting phase of the wave The speed v of the wave is: v = f λ = ω k
NOTE: both k and ω are positive, the signs in front of these numbers indicate the direction of the wave: if the signs are opposite (i.e -,+ or +,-) then the wave is travelling in the +x direction if the signs are the same ( +,+ or -,-) then the wave is travelling in the x direction From maths you should remember cos(x) = sin(x + π 2 ), so: cos (kx ωt + φ) = sin (kx ωt + (φ + π ) 2 ) It is therefore possible to rewrite a wave equation, using cos instead of sin, provided that we also change the phase φ. As already mentioned, the exact value of φ is usually not very important.
A travelling wave is described by the following equation: y(x, t) = 0.25 sin(0.5πx 4πt) For this wave find the: (a) amplitude (b) wave number (c) angular frequency (d) wavelength (e) frequency (f) wave speed (g) direction of travel (h) displacement at x = 1.3 m and t = 4.2 s Compare the given equation with the general form: y = 0.25 sin(0.5πx 4πt) y = A sin(kx ωt + φ) (a) A = 0.25 m (b) k = 0.5π = 1.57 m 1 (c) ω = 4π = 12.57 Hz
(d) λ = 2π k = 2π 0.5π = 4.0 m (e) f = ω 2π = 4π 2π = 2.0 Hz (f) v = ω k = 4π 0.5π = 8.0 m/s or v = f λ = (2)(4) = 8 m/s (g) There are opposite signs in front of k and ω, so the wave is travelling in the +x direction (h) y(x = 1.3, t = 4.2) = 0.25 sin(0.5π 1.3 4π 4.2) = 0.25 sin( 50.74 RADIANS) = 0.11 m
A travelling wave is described by the following equation: y(x, t) = 0.25 sin π(0.5x 4t) For this wave find the: (a) amplitude (b) wave number (c) angular frequency (d) wavelength (e) frequency (f) wave speed (g) direction of travel (h) displacement at x = 1.3 m and t = 4.2 s y(x, t) = 0.25 sin π(0.5x 4t) = 0.25 sin(0.5πx 4πt) This is the SAME equation as we just saw! Don t be fooled by our pitiful attempts to confuse you!!! Obviously all answers are the same!
A travelling wave is described by the following equation: y(x, t) = 0.25 cos π(0.5x 4t) For this wave find the: (a) amplitude (b) wave number (c) angular frequency (d) wavelength (e) frequency (f) wave speed (g) direction of travel (h) displacement at x = 1.3 m and t = 4.2 s This is the same equation as the previous example except with cos instead of sin. The answers (a) - (g) are unchanged, only (h) requires re-doing: y(x = 2, t = 4) = 0.25 cos π(0.5 1.3 4 4.2) = 0.25 cos( 50.74) again RADIANS = 0.22 m
A travelling wave is described by the following equation: y(x, t) = 0.25 cos π(4t 0.5x) For this wave find the: (a) amplitude (b) wave number (c) angular frequency (d) wavelength (e) frequency (f) wave speed (g) direction of travel (h) displacement at x = 1.3 m and t = 4.2 s Remember cos( X) = cos X, so: y(x, t) = 0.25 cos π(4t 0.5x) = 0.25 cos π(0.5x 4t) This is the same equation as the previous example and is another attempt to confuse you. The x term does not have to come first!
Transverse & Longitudinal Waves Contents There are two common types of travelling waves, differing in the exact motions they cause: Longitudinal waves have the wave vibrations in the same direction as the direction of the wave. Sound is a longitudinal wave, consisting of pressure variations in the air. Transverse waves are more common, including waves on a string, ocean waves and light. In these waves, the direction of the medium vibrations is transverse to (across, or perpendicular) the wave velocity. In both cases, the average motion of the medium is zero - particles oscillate about their original position. The definitions of wavelength, frequency apply equally to both. We ll now look a little more closely at one example of a transverse wave - waves on a string.
Wave on String: Transverse Velocity Contents Imagine an ant holding on to a string at position x as a harmonic wave passes by. The transverse displacement (i.e. the displacement in a direction transverse to the waves direction of motion - up/down) of the ant s point on the string at time t is: y(x, t) = A sin (kx ωt + φ) We know that the wave speed is v = ω k, but what is the speed of the ant? The ant is moving up and down with variable transverse velocity, v t where: v t (x, t) = y = ωa cos (kx ωt + φ) t (where we use t - the partial derivative with respect to time, which requires x to be treated as a constant)
Remember, this speed is unrelated to the speed of the wave. It is the speed of the up/down motion of the string (and therefore the ant) at x. Notes: The maximum transverse speed is ωa, which occurs when cos(...) = ±1, i.e. when sin(...) = 0 - the undisturbed position of the string. The minimum transverse speed is 0, which occurs when cos(...) = 0, i.e. when sin(...) = ±1 - the maximum amplitude of the movement. This transverse velocity is different in magnitude and direction to the wave velocity. The ant (and each piece of the string) is in simple harmonic motion. ( Remember y = A sin(ωt + constant) ) These motions are all slightly out of sync, as the value of the constant changes.
Wave on String: Transverse Acceleration Contents Clearly, the transverse velocity is changing, so we know that there must also be a transverse acceleration. Of course the transverse acceleration is the (partial) time derivative of the transverse velocity: Notes: a t (x, t) = v t t = ω2 A sin (kx ωt + φ) = ω 2 y The maximum transverse acceleration is ω 2 A, which occurs when y is also maximum. (just like simple harmonic motion) The minimum transverse acceleration is 0, when y = 0.
A travelling wave is described by the following equation: For this wave find the maximum y(x, t) = 0.25 sin π(0.5x 4t) (a) transverse speed (b) transverse acceleration (a) max. transverse speed is ωa = 4π 0.25 = 3.1 m/s. We previously found the speed of this wave to be 8 m/s, so the wave is faster than the transverse motion of the string. For a different wave, this could be the other way around. (b) max. transverse acceleration is ω 2 A = (4π) 2 0.25 = 39.5 m/s 2 4g
Wave on String: Wave Speed Contents Two important properties of a stretched string are: the mass per unit length (or linear mass density) : µ the tension T in the string We should expect the values of these properties to affect the properties of a wave on the string. To investigate this connection, let s consider wave travelling on a string at speed v to the right. Now, we zoom in on a small segment which is momentarily at rest, at the peak of the wave. v A
If we now consider ourselves to be in a frame of reference moving with the same velocity as the wave i.e. v to the right, then the velocity of the small segment relative to us is v to the left. The acceleration of the segment is the transverse acceleration seen earlier, directed down. We can draw the free body diagram of the forces acing on the segment as seen at right, where T 1 and T 2 are tensions, and m is the mass of the segment. v θ mg a θ T2 T 2 Assuming that the weight of the segment is negligible compared to the tensions, then T 1 = T 2 = T.
Applying Newton s Second Law: F = ma = 2T sin θ down The acceleration of the segment is perpendicular to the velocity, so the motion can be considered to be circular, with a connected to the speed v by a = v 2 R for an unknown radius R. 2T sin θ = ma = m v 2 R 2T sin θ = (µ 2 Rθ) v 2 θt = µ θv 2 T v = µ R (m = µ segment length) (sin θ θ for small angles) θ T v a θ θ R T θ
Any wave of any frequency travelling on this string must have this speed. Other waves also have this property, for instance sound waves of all frequencies will travel at the same speed in air. This speed is determined by the physical properties of the air - pressure, density etc. For some waves, the speed in a material will depend on the frequency. Probably the best-known example of this light travelling through glass or water. The differing speed of different wavelengths (i.e. colors) causes a rainbow formed by a prism, or by moisture in the air. A material like this, where speed depends on frequency of the wave is known as a dispersive medium and won t be discussed any further in this course.
Waves on String: Power Transmitted Contents The particles of the string do not move in the direction of the wave, but energy is transmitted along the string by the wave. The rate at which this occurs is the power carried by the wave. Let s look again at a small segment of the string we ve just dealt with. As it moves up and down, the energy of this small segment of the string is converted between potential and kinetic energies (just like a mass on a spring). When the displacement is zero, then the energy is all kinetic energy, where the speed is the maximum transverse speed v t = ωa. v v t
If the length of our small segment is taken to be x then its mass will be µ x and so its kinetic energy is: KE = 1 2 mv 2 t = 1 2 (µ x)(ωa)2 If the time taken for this energy to be transfered is t, then the power P carried by the wave is: P = E t = 1 ( ) x 2 µ ω 2 A 2 = 1 t 2 µvω2 A 2 This particular formula only applies to waves on a string, but similar formulas can be found for other waves, all with the common factor that: P ω 2 A 2
A string with linear mass density 5.0 10 2 kg/m is stretched with a tension of 80 N. A harmonic wave with amplitude 6 cm and frequency 60 Hz travels along the string. Find: (a) the speed of the wave (b) the power transferred by the wave down the string. (a) v = T µ = 80 5.0 10 2 = 40 m/s (b) P = 1 2 µvω2 A 2 = 1 2 (5.0 10 2 )(40)(2π 60) 2 (6 10 2 ) 2 = 512 W
Waves on a String Summary Contents As well as the definitions common to all waves for wavelength, frequency etc., there are some extra equations that apply only for a wave on a string: The speed v of the wave is determined by the physical properties of the string: T v = µ where T is the tension in the string, and µ is the linear mass density or mass per length of the string. The power P, carried by the wave is: P = 1 2 µvω2 A 2