Chapter 15: Mechanical Waves In Chapter 14, we talked about “wave-like” motions – oscillations – that were a function of time: x ( t ) = A cos (ωt + φ ) (mass on spring, e.g.) In this chapter, we discuss waves that depend on position also – waves that propagate (travel through some distance). Examples: • waves on strings • seismic waves • sound waves • electromagnetic waves ¾ We’ll focus for now on waves on strings. We will talk about sound waves in Chapter 16.

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Ch. 15: Mechanical Waves

Types of Mechanical Waves • transverse (waves on strings): motion of individual particles of the medium is perpendicular to (transverse to) the direction of propagation of the wave. • longitudinal (sound waves): motion of individual particles is in the same direction as the direction of propagation of the wave. • combination (water waves): motion of individual particles is partly transverse and partly longitudinal.

Figure 1 2

Ch. 15: Mechanical Waves

Speed of Propagation, Wavelength, and Frequency Consider a sinusoidal wave on a string:

Figure 2 3

Ch. 15: Mechanical Waves

The speed of propagation, v , is the speed with which the wave moves through the wave medium. The wavelength, λ , of the wave is the distance corresponding to one complete cycle of the wave (peak to peak, trough to trough, etc.). The frequency, f , is the number of complete cycles that occur per second. These three parameters are related by:

v=λf

(1)

To see why, consider what happens as the wave moves a distance equal to its wavelength. The time it takes for this to happen is the period, T . The speed of propagation is therefore:

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Ch. 15: Mechanical Waves

v=

dist λ ⎛1⎞ = = λ⎜ ⎟ time T ⎝T ⎠

But, from Ch. 14, we know:

1 = f T So:

v=λf

In general, the speed of propagation depends on physical properties of the wave medium.

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Ch. 15: Mechanical Waves

The Wave Function Consider a sinusoidal wave on a string originally lying along the x axis:

Figure 3

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Ch. 15: Mechanical Waves

The left end of the string is driven up and down (along the y axis) periodically. The displacement of an individual particle (i.e., an individual bit of the string) will be a function of x and t : y = y ( x, t )

The wave function is the function y ( x, t ) that gives the displacement of any particle in the medium at any time.

From Fig. 3, the position of the left end of the string is given by a sinusoidal function of time: y ( x = 0, t ) = A cos (ωt ) = A cos ( 2π ft ) The wave disturbance travels from x = 0 to any other x with speed v (the speed of propagation). So: x x v= ⇒ t= t v The displacement of a particle at position x at any time t is therefore the same as the displacement of the particle at position x = 0 at the earlier time t − x v . So for a particle at any x , the displacement y is given by:

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Ch. 15: Mechanical Waves

⎡ ⎛ x ⎞⎤ y ( x, t ) = A cos ⎢ω ⎜ t − ⎟ ⎥ ⎣ ⎝ v ⎠⎦ or (rewriting a bit):

⎡ ⎛ x t ⎞⎤ y ( x, t ) = A cos ⎢ 2π ⎜ − ⎟ ⎥ ⎣ ⎝ λ T ⎠⎦ Defining the wave number, k , to be: 2π k≡ , we can write y ( x, t ) as:

λ

y ( x, t ) = A cos ( kx − ωt )

(wave traveling in +x direction)

(2) (3)

For a wave traveling in the − x direction, we could repeat the above analysis with x replaced by − x . The result would then be: y ( x, t ) = A cos ( kx + ωt )

(wave traveling in -x direction)

(4)

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Ch. 15: Mechanical Waves

The Wave Number The wave number, k, is the spatial analogue of the angular frequency ω . The wavelength λ is the number of meters per cycle: ⎛ # meters ⎞ λ =⎜ ⎟ cycle ⎝ ⎠ The reciprocal of this is: 1 ⎛ #cycles ⎞ =⎜ (∗ ) ⎟ λ ⎝ meter ⎠ This is the spatial analogue of the frequency f . Each cycle corresponds to the angle kx going through 2π radians. So if we multiply ( ∗ ) by 2π , we get the number of radians per meter that the angle kx goes through: ⎛ 1 ⎞ ⎛ #radians ⎞ 2π ⎜ ⎟ = ⎜ ⎟ ⎝ λ ⎠ ⎝ meter ⎠ This is the wave number k . Comparing this with ω = 2π T , we see that k is like the spatial equivalent of ω . 9

Ch. 15: Mechanical Waves

Speed of Propagation in Terms of ω and k We know: But λ = 2π k and f = ω 2π , so:

v=λf

⎛ 2π ⎞⎛ ω ⎞ v=⎜ ⎟⎜ ⎟ ⎝ k ⎠⎝ 2π ⎠ v=

ω k

(5)

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Ch. 15: Mechanical Waves

Particle Velocity and Acceleration Differentiating (3) with respect to time gives the rate at which y is changing. This is the speed of a particle of the string as it executes its periodic, up-and-down motion: ∂ ∂ v y = ⎡⎣ y ( x, t ) ⎤⎦ = ⎡⎣ A cos ( kx − ωt ) ⎤⎦ ∂t ∂t Notice that we must take a partial derivative here because we want the rate of change of y at some fixed value of x (some particular point in the string). Using the chain rule,

∂ v y = ⎣⎡ − A sin ( kx − ωt ) ⎦⎤ [ −ωt ] ∂t

v y = ω A sin ( kx − ωt ) (6) Note that this is not the same thing as the speed of propagation! It’s the speed with which an individual particle moves up and down, transverse to the direction of propagation!

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Ch. 15: Mechanical Waves

The acceleration of a particle in the string as it undergoes its up-and-down motion is: ∂ 2 y ( x, t ) ay = = −ω 2 A cos ( kx − ωt ) 2 ∂t ∂ 2 y ( x, t ) 2 ω ay = = − y ( x, t ) 2 ∂t

(7) (8)

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Ch. 15: Mechanical Waves

The Wave Equation In a manner exactly similar to the process used to get (7), if I differentiate (3) twice with respect to x , I get: ∂ 2 y ( x, t ) 2 (9) = − k A cos ( kx − ωt ) 2 ∂x ∂ 2 y ( x, t ) 2 (10) = − k y ( x, t ) 2 ∂x Comparing (7) and (9), we see that: ∂ 2 y ( x, t ) ⎛ k 2 ⎞ ∂ 2 y ( x, t ) =⎜ 2 ⎟ 2 2 ∂x ⎝ ω ⎠ ∂t But from (5), k ω is the reciprocal of the speed of propagation: k 1 = ω v Therefore, we get: ∂ 2 y ( x, t ) ⎛ 1 ⎞ ∂ 2 y ( x, t ) (11) =⎜ 2 ⎟ 2 ∂x 2 v ∂ t ⎝ ⎠

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Ch. 15: Mechanical Waves

Eq. (11) is called the 1-D wave equation. Whenever a disturbance in any medium obeys a relation of the form shown in (11), we know that the disturbance can be represented as a wave propagating along the x axis with speed v .

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Ch. 15: Mechanical Waves

Speed of a Wave on a String

Suppose a sinusoidal wave is propagating in a string in the + x direction. Consider a segment of the string, as shown in Figure 4.

Figure 4

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Ch. 15: Mechanical Waves

Let the equilibrium length of this segment (i.e., when no wave is propagating) be called Δx . Let the mass per unit length of the string be called μ . The mass of this segment is, then: ⎛ mass ⎞ m=⎜ ( length of this segment ) = μΔx ⎟ ⎝ unit length ⎠ The forces on this segmentG are asGshown in Fig. 4. The horizontal components of the forces F1 and F2 must be equal in magnitude, because there is no horizontal acceleration of this segment. Accordingly, these two horizontal components are both called F in Fig. 4. From Fig. 4, we see that F2 y F is the slope of this segment at the right end. So: F2 y ⎛ ∂y ⎞ =⎜ ⎟ F ⎝ ∂x ⎠ x +Δx In a similar way,

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Ch. 15: Mechanical Waves

⎛ ∂y ⎞ = −⎜ ⎟ F ⎝ ∂x ⎠ x The net force in the y direction is therefore: ⎡⎛ ∂y ⎞ ⎛ ∂y ⎞ ⎤ ∑ Fy = F2 y + F1 y = F ⎢⎜⎝ ∂x ⎟⎠ − ⎜⎝ ∂x ⎟⎠ ⎥ x +Δx x⎦ ⎣ nd From Newton’s 2 law, then, ∂2 y ∑ Fy = ma y = m ∂t 2 F1y

⎡⎛ ∂y ⎞ ∂2 y ⎛ ∂y ⎞ ⎤ F ⎢⎜ ⎟ − ⎜ ⎟ ⎥ = ( μΔx ) 2 ∂t ⎣⎝ ∂x ⎠ x +Δx ⎝ ∂x ⎠ x ⎦ ⎡⎛ ∂y ⎞ ⎛ ∂y ⎞ ⎤ −⎜ ⎟ ⎥ ⎢⎜ ∂x ⎟ 2 ⎣⎝ ⎠ x +Δx ⎝ ∂x ⎠ x ⎦ = ⎛ μ ⎞ ∂ y ⎜ ⎟ 2 Δx ⎝ F ⎠ ∂t In the limit as Δx → 0 , the quantity on the LHS is the second partial derivative of y with respect to x :

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Ch. 15: Mechanical Waves

∂2 y ⎛ μ ⎞ ∂2 y (∗ ) =⎜ ⎟ 2 2 ∂x ⎝ F ⎠ ∂t Comparing ( ∗ ) with the wave equation (Eq. (11)), we can read off the speed of propagation in terms of the density of the string and the tension in the string: F v= (12)

μ

This says that the wave will propagate rapidly when the tension F is high and when the density (mass per unit length) of the string is low.

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Ch. 15: Mechanical Waves

Interference, Boundary Conditions, and the Principle of Superposition When a wave strikes the boundaries of its medium, all or part of the wave is reflected. The wave coming in toward the boundary is called the incident wave. In general, the reflected wave overlaps with the incident wave. This overlapping is called interference. When a wave is incident upon a boundary, the reflected wave can cause a displacement in the same direction as the incident wave or in the opposite direction. That is, the reflected wave can be “right-side-up” or “upside-down.” Which one of these occurs depends on the conditions at the boundary. For waves on a string, for example, the end of the string could be rigidly held in place or it could be free to move up and down. The conditions at the ends of the string are called boundary conditions. Figure 5 shows what happens to a pulse on a string when the end of the string is fixed or free to slide in a direction transverse to the direction of propagation.

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Ch. 15: Mechanical Waves

Figure 5

When the end of the string is fixed, the reflected pulse is inverted (“upside-down); when the end of the string is free to slide, the reflected pulse is in the same direction as the incident pulse (“right-side-up”). 20

Ch. 15: Mechanical Waves

The Principle of Superposition Consider two waves traveling on a string. Let the transverse displacement of the string caused by one of the waves be described by a wave function y1 ( x, t ) ; the displacement of the string caused by the other wave will be called y2 ( x, t ) .

When these two waves interfere, the actual displacement of any point on the string at any time is obtained by adding the displacement the point would have if only the first wave were present and the displacement it would have if only the second wave were present. This is called the principle of superposition. Mathematically, the principle of superposition says the net displacement of any bit of the string, y ( x, t ) , is given by:

y ( x, t ) = y1 ( x, t ) + y2 ( x, t ) (13) Figure 6 shows the net displacement of a string when two pulses of identical shape interfere.

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Ch. 15: Mechanical Waves

(a)

Figure 6

(b) 22

Ch. 15: Mechanical Waves

Standing Waves on a String Suppose we make a sinusoidal wave on a string of length L by driving the right end up and down continuously. This will produce a wave moving to the left which will be reflected at the left end. Because the left end is held fixed, the reflected wave will be inverted. We can choose initial conditions so that we can express the incident and reflected waves as cosines. Let’s call the incident wave y1 ( x, t ) and the reflected wave

y2 ( x, t ) . The wave functions would be:

y1 ( x, t ) = − A cos ( kx + ωt )

y2 ( x, t ) = + A cos ( kx − ωt ) By the principle of superposition, these two waves will interfere to produce a wave: y ( x, t ) = y1 ( x, t ) + y2 ( x, t ) = − A cos ( kx + ωt ) + A cos ( kx − ωt ) , or, making use of a trig. identity: y ( x, t ) = ⎡⎣ 2 A sin ( kx ) ⎤⎦ sin (ωt )

(14) (15)

(16)

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Ch. 15: Mechanical Waves

Figure 7

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Ch. 15: Mechanical Waves

In general, the wave y ( x, t ) does not produce a stationary pattern. However, for some frequencies, we get a stationary pattern of points where the amplitude is zero, called nodes, and points where the amplitude is a maximum, called antinodes. (The nodes are labeled N in Figure 7, while the antinodes are labeled A.) Such a wave pattern is called a standing wave. At the nodes, the waves y1 ( x, t ) and y2 ( x, t ) cancel exactly. This is called destructive interference. At the antinodes, the two peaks line up to give a peak of twice the amplitude of either wave. This is called constructive interference. The condition required in order to get a node is that y ( x, t ) must be zero for all t. From Eq. (16), this implies: 2 A sin ( kx ) = 0 or λ 3λ x = 0, , λ , ,... (17) 2 2 up to x = L . 25

Ch. 15: Mechanical Waves

As Figure 7 shows, successive nodes are half a wavelength apart, as are successive antinodes. The two sine factors in (16) oscillate between ± 1. Therefore, the amplitude of the standing wave is: ASW = 2 A and we can write: y ( x, t ) = ⎡⎣ ASW sin ( kx ) ⎤⎦ sin (ωt )

(18)

Harmonics Equation (17) implies that in order for standing waves to be produced, the length of the string, L , must be a multiple of λ 2 : λ 3λ L = , λ , ,... 2 2

λ

(19) L = n , n = 1, 2,3,... (string fixed at both ends) 2 (Note that we can’t have L = 0 because that would mean there is no string at all!)

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Ch. 15: Mechanical Waves

The first few of these cases are shown in Figure 8.

Figure 8

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Ch. 15: Mechanical Waves

Rearranging (19), we get a series of possible wavelengths for standing waves: 2L , n = 1, 2,3,... (string fixed at both ends) λn = (20) n Using v = λ f , this means there is a series of possible frequencies that will give standing waves: v f n = n , n = 1, 2,3,... (string fixed at both ends) (21) 2L These frequencies are called harmonics. The first frequency, corresponding to n = 1, is called the fundamental frequency, or simply the fundamental. The n = 2 frequency is called the second harmonic, or, in musical terminology, the first overtone (i.e., the first harmonic over the fundamental), and so on. When you play the “A” above “middle C” on the piano, the fundamental frequency is f1 = 440 Hz. This is primarily the tone you hear, but there are higher harmonics present in different amounts (i.e., with different amplitudes). The relative amplitudes of the harmonics give rise to what

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Ch. 15: Mechanical Waves

musicians call timbre. Different instruments producing sounds with the same fundamental sound different because of their spectral content. Using v = F μ for the speed of a wave on a string, we can write (21) as: n F , n = 1, 2,3,... (string fixed at both ends) (22) fn = 2L μ

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