Question

A wave is modeled with the function y ( x , t ) = 0 . 2 5 cos ( 0 . 3 0 x − 0 . 9 0 t + π/ 3 ) where all lengths are in meters and all times in seconds.

a. Find the wavelength of the wave.

b. Find the period of the wave.

c. Find the wave speed (a positive number).

d. What is the instantaneous velocity of one of the particles that make up the medium at position 0 at time 22 sec? Make sure to use the correct sign (plus or minus) for your answer.

(Also can you help me with this question? I know its simple but I keep getting it wrong)

Two car horns are sounded creating two sound waves with frequencies that differ by a factor of three. How does the speed of the higher frequency sound wave compare to the lower frequency sound wave?

Answer 1

The wave function of this problem can be written as follows:

$y(x,t)=Acos(kx-wt+\phi )\Rightarrow y(x,t)=0.25cos(0.30x-0.90t+\pi /3)$

where

$A=0.25$

$k=0.30$

$w=0.90$

$\phi =\pi /3$

a. Find the wavelength of the wave.

The wavelength is given by the following equation:

$\lambda =\frac{2\pi}{k}=\frac{2\pi}{0.30}$

$\lambda =20.94m$

b. Find the period of the wave.

The period is given by the following equation:

$T=\frac{2\pi}{w}=\frac{2\pi}{0.90}$

$T=6.98 s$

c. Find the wave speed

The speed is given by the following equation:

$v=\frac{\lambda }{T}=\frac{20.94m}{6.98s}$

$v=3m/s$

d. What is the instantaneous velocity of one of the particles that make up the medium at position 0 at time 22 sec?

To determine the velocity we will perform the derivative of the wave function:

$v=\frac{dy}{dt}=-(0.25)(0.90)sin(0.30x-0.90t+\pi /3)$

$v(x,t)=-(0.25)(0.90)sin(0.30x-0.90t+\pi /3)$

since the position is zero, then x = 0 and the equation is as follows:

$v(0,t)=-(0.25)(0.90)sin(-0.90t+\pi /3)$

and t=22 s

$v(0,22 s)=-(0.25)(0.90)sin(-0.90(22)+\pi /3)$

$v(0,22 s)=-0.066m/s$