The expression for the equation of a wave is:
[tex]y(x,t)= A \cos (kx + \omega t)[/tex] (1)
where
A is the amplitude
[tex]k= \frac{2 \pi}{ \lambda } [/tex] is the wave number, with [tex]\lambda[/tex] being the wavelength
x is the displacement
[tex]\omega= \frac{2 \pi}{T} [/tex] is the angular frequency, with T being the period
t is the time
The equation of the wave in our problem is
[tex]y(x,t)= 1.6 \cos (0.71 x + 36 t)[/tex] (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,
[tex]k= \frac{2 \pi}{\lambda}=0.71 cm^{-1} [/tex]
And re-arranging this relationship we find [tex]\lambda= \frac{2 \pi}{k}= \frac{2 \pi}{0.71 cm^{-1}}=8.8 cm [/tex]
c) Period:
we can find the period by using the angular frequency:
[tex]\omega= \frac{2 \pi}{T}= 36 s^{-1}[/tex]
By re-arranging this relationship, we find
[tex]T= \frac{2 \pi}{\omega}= \frac{2 \pi}{36 s^{-1}}=0.17 s [/tex]
d) Speed of the wave:
The speed of a wave is given by
[tex]v= \lambda f[/tex]
where f is the frequency of the wave, which is the reciprocal of the period:
[tex]f= \frac{1}{T}= \frac{1}{0.17 s}=5.9 s^{-1} [/tex]
And so the speed of the wave is
[tex]v= \lambda f=(8.8 cm)(5.9 s^{-1})=52 cm/s[/tex]
e) Direction of the wave:
A wave written in the cosine form as
[tex]y(x,t)=A \cos(\omega t- kx)[/tex]
propagates in the positive x-direction, while a wave written in the form
[tex]y(x,t)=A \cos(\omega t+ kx)[/tex]
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.
