Question

Find (a) the amplitude, (b) the wavelength, (c) the period, and (d) the speed of a wave whose displacement is given by y= 1.6 cos( 0.71 x+ 36 t), where x and y are in cm and t is in seconds. part a part complete express your answer using two significant figures. a = 1.6 cm previous answers correct part b part complete express your answer using two significant figures. λ = 8.8 cm previous answers correct part c express your answer using two significant figures. t = s request answer part d express your answer using two significant figures. v = cm/s request answer part e in which direction is the wave propagating?

Answer 1

The expression for the equation of a wave is:
$y(x,t)= A \cos (kx + \omega t)$ (1)
where
A is the amplitude
$k= \frac{2 \pi}{ \lambda }$ is the wave number, with $\lambda$ being the wavelength
x is the displacement
$\omega= \frac{2 \pi}{T}$ is the angular frequency, with T being the period
t is the time

The equation of the wave in our problem is
$y(x,t)= 1.6 \cos (0.71 x + 36 t)$ (2)
where x and y are in cm and t is in seconds.


a) Amplitude:
if we compare (1) and (2), we immediately see that the amplitude of the wave is the factor before the cosine:
A=1.6 cm

b) Wavelength:
we can find the wavelength starting from the wave number. For the wave of the problem, 
$k= \frac{2 \pi}{\lambda}=0.71 cm^{-1}$
And re-arranging this relationship we find $\lambda= \frac{2 \pi}{k}= \frac{2 \pi}{0.71 cm^{-1}}=8.8 cm$

c) Period:
we can find the period by using the angular frequency:
$\omega= \frac{2 \pi}{T}= 36 s^{-1}$
By re-arranging this relationship, we find
$T= \frac{2 \pi}{\omega}= \frac{2 \pi}{36 s^{-1}}=0.17 s$

d) Speed of the wave:
The speed of a wave is given by
$v= \lambda f$
where f is the frequency of the wave, which is the reciprocal of the period:
$f= \frac{1}{T}= \frac{1}{0.17 s}=5.9 s^{-1}$
And so the speed of the wave is
$v= \lambda f=(8.8 cm)(5.9 s^{-1})=52 cm/s$

e) Direction of the wave:
A wave written in the cosine form as
$y(x,t)=A \cos(\omega t- kx)$
propagates in the positive x-direction, while a wave written in the form
$y(x,t)=A \cos(\omega t+ kx)$
propagates in the negative x-direction. By looking at (2), we see we are in the second case, so our wave propagates in the negative x-direction.