Chapter 15 Outline Mechanical Waves Mechanical wave types Periodic waves Sinusoidal wave Wave equation Energy and intensity Superposition of waves Standing waves on strings Normal modes Stringed instruments Resonance

Mechanical Wave A disturbance that travels through some medium is a mechanical wave. Sound, ocean waves, vibrations on strings, seismic waves… In each case, the medium moves from and returns to its equilibrium state. The wave transports energy, not matter. The disturbance propagates with a wave speed that is not the same as the speed at which the medium moves.

Transverse Wave If the motion of the medium is perpendicular to the propagation of the wave, it is a transverse wave. e.g. wave on a string

Longitudinal Wave If the motion of the medium is along the propagation of the wave, it is a longitudinal wave. e.g. sound waves We can also have a combination of the two.

Periodic Waves If we shake a stretched string once, a wave pulse will travel along the string, but afterwards the string returns to its flat equilibrium position. If instead, we attach the string to a simple harmonic oscillator, we will produce a periodic wave. A simple example is a sinusoidal wave.

Transverse Sinusoidal Waves As we discussed last chapter, SHM is described by its amplitude, 𝐴, and period, 𝑇, (as well as the corresponding frequency, 𝑓=1/𝑇, and angular frequency, 𝜔=2𝜋𝑓. Over one period, the wave advances a distance we call the wavelength, 𝜆. Distance for one full wave pattern. The wave speed (distance divided by time) is therefore 𝑣=𝜆/𝑇. It is more commonly written in terms of the frequency. 𝑣=𝜆𝑓

Longitudinal Sinusoidal Waves In a longitudinal wave, the medium oscillates along the direction of propagation. A common example is a sound wave. The wave is composed of compressions (high density) and rarefactions (low density).

Wave Function As in the harmonic motion from last chapter, we want to mathematically describe the displacement of the medium. Since the wave is moving, the displacement depends on both the position and the time. 𝑦=𝑦(𝑥,𝑡) Consider the case of a sinusoidal wave with amplitude 𝐴 and angular frequency 𝜔, propagating at speed 𝑣.

Wave Function So, the wave can be described by: 𝑦 𝑥,𝑡 =𝐴 cos (𝑘𝑥−𝜔𝑡) The wave number, 𝑘, is defined as: 𝑘= 2𝜋 𝜆 Since 𝜔=2𝜋𝑓, the wave velocity is: 𝑣=𝜆𝑓= 2𝜋 𝑘 𝜔 2𝜋 𝑣= 𝜔 𝑓 Note: This is referred to as the phase velocity.

The Wave Equation We have described sinusoidal waves, but we can derive an expression that will hold for any periodic wave. Starting with the sinusoidal wave function, 𝑦 𝑥,𝑡 =𝐴 cos (𝑘𝑥−𝜔𝑡) Taking partial derivatives with respect to position and time, we find that 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑥 2 = 1 𝑣 2 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑡 2 This is called the wave equation, and it is true of any periodic wave, even non-mechanical waves (light).

Speed of a Wave on a String Consider a small segment of the string, length ∆𝑥, mass 𝑚, with a linear mass density 𝜇=𝑚/∆𝑥. Normally we use lambda for linear density, but we are using it already for wavelength. The forces acting on either end of the segment, 𝑭 1 and 𝑭 2 , pull along the string. (At rest, these would just be the tension, and would cancel each other.) Because the string is displaced from equilibrium, these forces are not parallel. The horizontal components cancel, but the vertical components lead to the transverse acceleration.

Speed of a Wave on a String By comparing the forces to the slope of the string at that point, we can relate the horizontal force to the transverse acceleration. Using the wave equation, we can then find the wave speed.

Speed of a Wave on a String After some manipulation, 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑥 2 = 𝜇 𝐹 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑡 2 The wave equation: 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑥 2 = 1 𝑣 2 𝜕 2 𝑦 𝑥,𝑡 𝜕 𝑡 2 So, the wave speed must be: 𝑣= 𝐹 𝜇 Since 𝐹 is the horizontal component of the force it must be equal to the tension, 𝑇. We are not using 𝑇 so as not to confuse it with the period.

Wave on a String Example

Energy in Wave Motion As we discussed at the beginning of class, waves do not transport matter, but they do transport energy. Recall that power (𝑃=𝑑𝑊/𝑑𝑡) can be expressed as 𝑃=𝐹𝑣 Consider a small segment of the string. It moves up and down because the wave is doing work on it. Only the vertical component of the force does work, and the velocity is also vertical. 𝐹 𝑦 =−𝐹 𝜕𝑦 𝜕𝑥 𝑣 𝑦 = 𝜕𝑦 𝜕𝑡 Combining these, 𝑃=−𝐹 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑡

Energy in Wave Motion For a sinusoidal wave, 𝑦 𝑥,𝑡 =𝐴 cos (𝑘𝑥−𝜔𝑡) So the power is 𝑃=−𝐹 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑡 =−𝐹[−𝑘𝐴 sin 𝑘𝑥−𝜔𝑡 ][𝜔𝐴 sin 𝑘𝑥−𝜔𝑡 ] 𝑃=𝐹𝑘𝜔 𝐴 2 sin 2 𝑘𝑥−𝜔𝑡 Combining this with 𝑘=𝜔/𝑣 and 𝑣= 𝐹/𝜇 𝑃= 𝜇𝐹 𝜔 2 𝐴 2 sin 2 𝑘𝑥−𝜔𝑡

Maximum and Average Power Since sine of any angle cannot exceed one, the maximum power is: 𝑃 max = 𝜇𝐹 𝜔 2 𝐴 2 The average power is then 𝑃 ave = 1 2 𝜇𝐹 𝜔 2 𝐴 2

Wave Intensity A wave on a string only transports energy in one dimension, but many other waves propagate in three dimensions. If this propagation is uniform, the power is spread out over the surface of a sphere, and we define the intensity, 𝐼, to be the average power transported per unit area. 𝐼= 𝑃 𝐴 =𝑃/4𝜋 𝑟 2 This means that the intensity decreases with 1/ 𝑟 2 . 𝐼 1 𝐼 2 = 𝑟 2 2 𝑟 1 2

Wave Power Example