HW 11 is due tomorrow night. Lecture 29 Goals: Chapter 20, Waves No labs this week. HW 11 is due tomorrow night. HW 12 (a short one) is due Thursday night. 1

Relationship between wavelength and period D(x,t=0) x x0 λ Alright, so we had started our discussion of waves. We argued that waves are disturbances that propagate through space. We discussed that you really need a two dimensional function to describe a wave, displacement as a function of x and t. At a particular time, such as t=0 here, we sort of look at a snapshot of a wave, where we observe displacement as a function of x as shown here. Or, we can choose to look at the displacement as a function of time at some fixed position x0. If we did that we would sort of look at the history graph, we would be plotting the displacement as a function of time. Now, we discussed sinusoidal waves.These are waves where the displacement is a sinusoidal function of both space and time. They are idealizations, so they are assumed to extend infinitely in both directions. Now, for a sinusoidal wave, the spatial period in space, which is the wavelength, is related to the temporal period through the speed of the wave. So the period is the wavelength divided by the speed. T=λ/v

Mathematical formalism D(x=0,t) D(x=0,t) ~ A cos (wt + f) w: angular frequency w=2π/T t T D(x,t=0) x λ D(x,t=0) ~ A cos (kx+ f) k: wave number k=2π/λ OK, so the displacement will be a two dimensional function, now we will discuss what this function is going to be for a sinusoidal wave. In this discussion, we will borrow a lot of the concepts from our discussion of simple harmonic motion. So, let’s first argue that we already know what this function should be at fixed position, or at fixed time. So once again, at some fixed position, we know that the history graph looks like an oscillation as a function of time. We already know, how to describe such an oscillation. While we were discussing SHM, we argued that we can represent such a sinusoidal function as A coswt, where A is the amplitude of the wave, phi is the phase constant, and w is the angular frequency w=2pi/T. So at some fixed position in space, we would expect this two dimensional function to reduce to something of this form. 3) Now, very similarly, a snapshot of the wave at some point in time is also a sinusoidal function, but now a sinusoidal function of space. Now, in perfect analogy with this form here, we can represent this function as A cos (kx+phi) where A is again the amplitude, but now k is 2pi/lambda where lambda is the spatial period. So if we take a snapshot of the wave at some point in time, we would expect the displacement function to reduce to something of this form. 4) Now, because T and Lmabda are related by the speed of the wave, k and omega are also related by the speed of the wave.

Mathematical formalism The two dimensional displacement function for a sinusoidal wave traveling along +x direction: D(x,t) = A cos (kx - wt + f) A: Amplitude k: wave number ω : angular frequency ϕ: phase constant 1) So we are looking for a function that reduces to cos(wt) at fixed point in distance, and cos(kx) at a fixed point in time, so it is not that difficult to see that for a wave propagating in the positive x direction, the proper two dimensional displacement function is going to be D(x,t)= A cos(kx-wt+phi), where once again, A is the amplitude of the wave, k is the wave number, w is the angular frequency and phi is phase constant. Now, we are going to discuss why there is a – sign here, but let’s observe that this form satisfies what we were looking for. At some fixed position, the term kx becomes a constant and this becomes a sinusoidal function in time. At some fixed position in time, this becomes a constant and we have a sinusoidal variation in space.

Mathematical formalism Note that there are equivalent ways of describing a wave propagating in +x direction: D(x,t) = A cos (kx - wt + f) D(x,t) = A sin (kx - wt + f+π/2) D(x,t) = A cos [k(x – vt) + f] Now, there are equivalent ways of describing this function, let’s quickly go over these. First of all, this function can be equivalently described by a sine instead of a cosine. We only need to add a pi/2 here, and these two functions would be equivalent. Also, this function can be written in a slightly different form. If we take k out of the paranthesis here, since k and w are related by the speed of the wave, the displacement can be written as k times x-vt. Argue arbitrary displacement functions f(x-vt)

Why the minus sign? As time progresses, we need the disturbance to move towards +x: at t=0, D(x,t=0) = A cos [k(x-0) + f] at t=t0, D(x,t=t0) = A cos [k(x-vt0) + f] vt0 v Now let’s discuss where the minus sign is coming from. So the wave is traveling in the +x direction, to the right, which means that as time progresses, we need the disturbance, the whole function move to the right. So let’s say that at t=0, we have this snapshot, and for the function, we plug in a zero for t, which means that we get a zero here. Now at a later time t0, instead of a 0, this term becomes vt0, and because of this minus sign, the whole structure moves to the right, because you need larger x values to produce equivalent numbers to the previous expression. x

Which of the following equations describe a wave propagating towards -x: D(x,t) = A cos (kx – wt ) D(x,t) = A sin (kx – wt ) C) D(x,t) = A cos (-kx + wt ) D) D(x,t) = A cos (kx + wt )

Tstring: tension in the string μ=M/L : mass per unit length Speed of waves The speed of mechanical waves depend on the elastic and inertial properties of the medium. For a string, the speed of the wave can be shown to be: Tstring: tension in the string μ=M/L : mass per unit length Now let’s discuss the speed of the waves a little bit. For mechanical waves, the speed of the waves will depend on the elastic and inertial properties of the medium. So for example, when you create a disturbance in a string, if you displace the particles from their equilibrium position, the speed will depend on how fast the particles get back to their equilibrium position. If the particles have large inertia, large mass, it will take a long time for the particles to get back to their initial position, and therefore the wave would be slower. Now, for a string ….

Waves on a string Making the tension bigger increases the speed. Making the string heavier decreases the speed. The speed depends only on the nature of the medium, not on amplitude, frequency etc of the wave.

(a) increase (b) decrease (c) stay the same Exercise Wave Motion A heavy rope hangs from the ceiling, and a small amplitude transverse wave is started by jiggling the rope at the bottom. As the wave travels up the rope, its speed will: v (a) increase (b) decrease (c) stay the same

Sound, A special kind of longitudinal wave λ Now, although there are many different kinds of waves in nature, two are especially important for us light waves and sound waves. As I mentioned last time, light waves are basically transverse electromagnetic waves with electric and magnetic fields orthogonal to the direction of propagation. Now, sound waves in air are a special kind of longitudinal wave. Although we usually think of sound waves to be in air, you can generate sound waves in any gas, in liquids and in solids. So these are longitudinal waves which means that the particle motion is in the same direction as the wave propagation. So let’s say that we are generating a sound wave using a vibrating loudspeaker. Each time the loudspeaker membrane moves forward, it collides with the air molecules and pushes them closer to each other. When the membrane moves backward, the fluid has room to expand and the density decreases a little. As a result, regions of higher and lower density gas and therefore higher and lower pressure gas is produced. An exaggerated picture is shown here. And this pattern moves away from the speaker. The speed of sound in air is about 300 m/s and actualy it does have some temperature dependence. As the temperature of air increases, the speed will also increase. In solids, the sound speeds are typically an order of magnitude, reaching values as high as 7-8 km/sec. Individual molecules undergo harmonic motion with displacement in same direction as wave motion.

Waves in two and three dimensions Waves on the surface of water: circular waves wavefront 1) Mention spherical waves

Plane waves Note that a small portion of a spherical wave front is well represented as a “plane wave”. 1) A good example is ocean waves

Intensity (power per unit area) A wave can be made more “intense” by focusing to a smaller area. I=P/A : J/(s m2) R 1) The concept of intensity quantifies the notion that as you move away from the source of the wave then the strength of the decreases. 1) Give the example of the sun.

Exercise Spherical Waves You are standing 10 m away from a very loud, small speaker. The noise hurts your ears. In order to reduce the intensity to 1/4 its original value, how far away do you need to stand? (A) 14 m (B) 20 m (C) 30 m (D) 40 m

I0: threshold of human hearing I0=10-12 W/m2 Intensity of sounds The range of intensities detectible by the human ear is very large It is convenient to use a logarithmic scale to determine the intensity level, b I0: threshold of human hearing I0=10-12 W/m2

Intensity of sounds Some examples (1 pascal  10-5 atm) : Sound Intensity Pressure Intensity (W/m2) Level (dB) Hearing threshold 3  10-5 10-12 Classroom 0.01 10-7 50 Indoor concert 30 1 120 Jet engine at 30 m 100 10 130 The dynamic range of the human ear is just incredible. The sensitivity of the ear drum is about an angstrom, so you can detect atomic scale motion that happens on your ear drum.

observer The Doppler effect The frequency of the wave that is observed depends on the relative speed between the observer and the source. observer

vs observer

f=f0/(1-vs/v) f=f0/(1+vs/v) The Doppler effect Approaching source: Receding source: f=f0/(1+vs/v)