Question

1. A wave on a string has the form y(x, t)=−20 cos(3pi x - 12pi t + pi/6), with all values in SI units.
a)Find its amplitude, (linear) frequency, wavelength, speed of propagation, direction of travel, maximum transverse speed and maximum transverse acceleration of the string. (Clearly label each of these answers and express in the appropriate units.)
b)Write the equation of another wave which if added to this one will produce a standing wave pattern. c)Provide the equation of the resulting standing wave.

2. In each case below, state whether the function represents a traveling wave or not:
a) y(x,t) = 3cos[pi(x/2 -t)]
b) y(x,t) = 3cos[pi(x/2 -t)] sin[pi(x/2 +t)]
c) y(x,t) = 3cos[pi(x/2 -t)] sin[pi(x-2t)]
d) y(x,t) = 3sin[pi(x/2 -t)] sin[pi(2x-t+4)]
e) y(x,t) = 3(e)^-(3x+2t)

Answer 1

1) a)

$$A=20 \ m$$

$$f=\frac{12\pi}{2\pi}=6\ Hz$$

$$k=\frac{3\pi}{2\pi}=1.5\ m^{-1}$$

So,

$$\lambda=\frac{2}{3}=0.67\ m$$

$$v=\lambda f=6\frac{2}{3}=4\frac{m}{s}$$

To find the direction of propogation of the wave, just look at the sign between the x and t terms in the equation. In our case we have a minus sign. A negative sign means the wave is traveling in the +x direction.

$$v_t=(12\pi)(20)=240\pi =754\frac{m}{s}$$

$$a_t=(12\pi)^2(20) =28424\frac{m}{s^2}$$

b)

$$y'(x, t)=−20 \cos(3\pi x +12\pi t +\frac{\pi}{6})$$

c)

$$y''(x,t)=-40 \cos(12\pi t)\sin{(3\pi x +\frac{\pi}{6})}$$

2) a) Yes.

b) No.

c)No.

d) No.

e) No.