Function for y(x,t) wave some questions

In summary, the function for the wave traveling along a string in the positive x-direction at 33 m/s with a frequency of 48 Hz is y(x,t) = Acos(9.139x - 301.59t + something). To find the amplitude and phase constant, take the time derivative of the function and solve for A and something using the given values for x=0, t=0. The correct values for A and something should be 0.003m and 295.31, respectively.
  • #1
yjk91
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Homework Statement


A wave travels along a string in the positive x-direction at 33 m/s. The frequency of the wave is 48 Hz. At x = 0 and t = 0, the wave velocity is 2.8 m/s and the vertical displacement is y = 3 mm. Write the function y(x, t) for the wave. (Use the following as necessary: x and t. Assume x and y are in m, and that t is in s.)


The Attempt at a Solution



y(x,t) = Acos(kx - wt)
should be the formula

i'm assuming A = 0.003m but it is not so i don't know

w = 2 * pi * f (frequency)
= 2 * pi * 48Hz
= 301.59

k = w / v
= 301.59 / 33(m/s)
= 9.139

so what i have now is y(x,t) = Acos(9.139x - 301.59t + something)

how do i find the amplitude and the missing something if there is one?

thank you
 
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  • #2
your vertical displacement y(x,t) would be 3mm since at t=0 and x=0 , y=3mm

y(x,t) = Acos( k(0) - w(0)) = A(1) = 3mm = 0.003m

what you have looks right to me. Sorry if its not to clear, it's been a long summer
 
  • #3
Liquidxlax said:
your vertical displacement y(x,t) would be 3mm since at t=0 and x=0 , y=3mm

y(x,t) = Acos( k(0) - w(0)) = A(1) = 3mm = 0.003m

what you have looks right to me. Sorry if its not to clear, it's been a long summer



one example was this question

A wave travels along a string in the positive x-direction at 31 m/s. The frequency of the wave is 47 Hz. At x = 0 and t = 0, the wave velocity is 2.8 m/s and the vertical displacement is y = 3 mm. Write the function y(x, t) for the wave. (Use the following as necessary: x and t. Assume x and y are in m, and that t is in s.)

the answer was 0.0099 * sin(9.53*x - 295.31*t + 2.84)

i know how to get the k and w which was 9.53 and 295.31
but i don't know how to get 0.0099 and 295.31 are
thank you

i know how to get the k and w which are 9.
 
  • #4
Find the velocity of the piece of string at x=0, t=0, by taking the time derivative of your function. Then there two equations for the unknown amplitude A and phase constant (something:smile:)

ehild
 
  • #5
for your help!

I would like to first acknowledge that you have made a good attempt at solving this problem. Your understanding of the formula for a wave is correct. However, there are a few things that need to be clarified in order to find the missing values in the function y(x,t).

First, the amplitude (A) in this case is not equal to the vertical displacement (y) at x = 0 and t = 0. The amplitude of a wave is the maximum displacement from the equilibrium position, which in this case is not given. Therefore, we cannot assume that A = 0.003 m.

To find the amplitude, we would need to know the maximum displacement of the wave from its equilibrium position. This information is not provided in the given question. However, we can use the given vertical displacement (y = 3 mm) and the formula y = A * cos(kx - wt) to find the amplitude.

Substituting the given values, we get:

3 mm = A * cos(0 - 0 + something)

Since cos(0) = 1, we can simplify the equation to:

3 mm = A * 1

Therefore, the amplitude (A) of the wave is 3 mm.

Now, to find the missing value in the function y(x,t), we can use the given wave velocity (v = 33 m/s) and the formula k = w/v, as you correctly did. However, we also need to know the value of w (angular frequency) in order to solve for the missing value in the function.

To find w, we can use the given frequency (f = 48 Hz) and the formula w = 2 * pi * f, as you also correctly did. Substituting the values, we get:

w = 2 * pi * 48 Hz
= 301.59 rad/s

Now that we have the values for w and k, we can substitute them into the function y(x,t):

y(x,t) = Acos(kx - wt)
= (3 mm) * cos(9.139x - 301.59t)

Therefore, the function for the wave is y(x,t) = (3 mm) * cos(9.139x - 301.59t).

I hope this explanation helps you understand how to find the missing values in the function y(x,t) for a wave
 

Related to Function for y(x,t) wave some questions

1. What is the role of x and t in the function y(x,t)?

The variables x and t in the function y(x,t) represent the position and time, respectively. They determine the shape and movement of the wave described by the function.

2. How is the amplitude of the wave related to the function y(x,t)?

The amplitude of the wave is represented by the value of y in the function y(x,t). It determines the maximum displacement of the wave from its equilibrium position.

3. Can you explain the concept of wavelength in relation to the function y(x,t)?

Wavelength is the distance between two consecutive points on a wave that are in phase, meaning they have the same amplitude and direction of motion. In the function y(x,t), it is represented by the distance between two consecutive peaks or troughs of the wave.

4. How does the frequency of the wave affect the function y(x,t)?

The frequency of the wave, which is the number of complete oscillations per unit time, is inversely proportional to the wavelength. This means that as the frequency increases, the wavelength decreases and the wave becomes more compressed. In the function y(x,t), the frequency is represented by the coefficient in front of the time variable (t).

5. What is the significance of the function y(x,t) in wave mechanics?

The function y(x,t) is a mathematical representation of a wave and is used to describe various physical phenomena such as sound, light, and water waves. It allows us to understand and predict the behavior of waves in different scenarios and is key to understanding many natural processes and technological applications.

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