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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?

User Redskull
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1 Answer

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The expression for the equation of a wave is:

y(x,t)= A \cos (kx + \omega t) (1)
where
A is the amplitude

k= (2 \pi)/( \lambda ) is the wave number, with
\lambda being the wavelength
x is the displacement

\omega= (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= (2 \pi)/(\lambda)=0.71 cm^(-1)
And re-arranging this relationship we find
\lambda= (2 \pi)/(k)= (2 \pi)/(0.71 cm^(-1))=8.8 cm

c) Period:
we can find the period by using the angular frequency:

\omega= (2 \pi)/(T)= 36 s^(-1)
By re-arranging this relationship, we find

T= (2 \pi)/(\omega)= (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= (1)/(T)= (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.
User Rads
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