Solving Wave Properties and Displacements: Transverse Waves on a String

[eku_girl83]

Here's my question (parts a-c are correct, but I need help with d and e):
Transverse waves on a string have a wave speed 8 m/s, amplitude .09 m, and a wavelength .38 m. The waves travel in the -x direction, and at t=0 the x=0 end of the string has zero displacement and is moving in the +y direction.
Calculator should be in radians
a) Find the following properties of these waves.
Frequency 21.05 Hz
Period .0475 seconds
Wave number 16.53 rad/m
b) Complete the wave function describing the wave
.09m*sin 2pi[(21.05 Hz)t+(2.63/m)x]
or (without units) = .09 Sin 2pi[21.05*t+2.63x]
c) Find the transverse displacement of a particle at x=.36m at time t=.15s
y=5.5702 cm
Everything is correct up to this point
Here's what I have Wrong:
d) How much time must elapse from the instant in part c until the particle at x=.36 m next has zero displacement?
e) How much additional time must elapse from the instant in part d until the next time the particle has zero displacement?

If anyone oculd help me, I would greatly appreciate it!
Thanks!

 

[turin]

d) How much time must elapse from the instant in part c until the particle at x=.36 m next has zero displacement?
e) How much additional time must elapse from the instant in part d until the next time the particle has zero displacement?

Both of these questions are two-fold execises: converting the phase of the wave to the time and using your knowledge of the zeros of sinusoids.

The phase of the wave is given by the argument of the sinusoid:

φ(x,t) = k x + ω t

where k is the wave number, x is the position of consideration, ω is the angular frequency, and t is the time of consideration. Perhaps the more familiar for of the phase is:

φ(x,t) = ( 2π / λ ) x + 2π f t,

where λ is the wavelength and f is the cyclic frequency.

You know the point on the string that you are considering, so that fixes x. Let ( 2π / λ ) x = θ (which you can calculate from the information that you have). Then, the phase is:

φ(x,t) = 2π f t + θ.

You can solve this for t. Then, use your knowledge of the zeros to decide what values of φ(x,t) will satisfy the required condition. The last requirement that you want the next zero, and not just any zero, fixes φ(x,t) to one of these values.

 

[eku_girl83]

If I set kx+wt=0 and solve for t when x=.36m, this still doesn't give me the correct answer... I'm a little confused on what you mean by "use your knowledge of zeroes of a sinusoid." Could you please clarify this? Thanks for the help!

 

[turin]

If I set kx+wt=0 and solve for t when x=.36m, this still doesn't give me the correct answer... I'm a little confused on what you mean by "use your knowledge of zeroes of a sinusoid." Could you please clarify this? Thanks for the help!

You know that you are looking for the next zero, which gives t > 0.15 s as an implicit requirement. Therefore, you know to throw out kx+wt=0, because that would require a t < 0.15 s (and time travel into the past is not allowed ). So, you know you can't use the zero: sin(0) = 0. You need to find the first zero that satisfies the implicit "next" condition. The same goes for part e. Your knowledge of zeros of a sinusoid should include that notion that they repeat periodically in the phase.

sin(φ(t)) = 0

such that

t is as small as possible but > 0.15 s for part d

and then increment the phase to the next zero for part e

 

[HallsofIvy]

sin(x) is 0 not just at x= 0 but also at x= &pi;, 2&pi;, etc.

Set x= .36 in kx+wt= &pi; and solve for t.