# Homework Help: Frequency and wavelength of a wave

1. Jan 23, 2005

### Shay10825

Hi everyone .

If I had a sine wave on a string at one instant of time how come when I reduce the wavelength and frequency by half the graph of the wave does not change.

~Thanks

2. Jan 23, 2005

### Gokul43201

Staff Emeritus
What are you plotting on the graph ?

3. Jan 23, 2005

### Shay10825

The question said :

The figure below shows a sine wave on a string at one instant of time (it is a normal sine wave). Which of the graphs below shows a wave where the wavelength and frequency asr each reduced by half? The answer was the same graph.

4. Jan 23, 2005

### futb0l

Well, the waveform will be t he same, it's just that the frequency is halfed. So that means the period of the graph is doubled.

5. Jan 23, 2005

### Shay10825

What does the change in wavelength do to the graph.

6. Jan 23, 2005

### Shay10825

If the period is doubled then why does the graph look the same as the original?

7. Jan 23, 2005

### christinono

What about the speed of the string?
You must look at the formula v=frequency*wavelength

8. Jan 23, 2005

### Curious3141

Are you certain that a change in the wavelength will not change the graph of the wave at an instant in time ?

I agree with you about the frequency, though. The instantaneous shape of the graph will not alter with frequency changes.

Do you know the one dimensional wave equation ?

$$a = a_0\cos(kx - \omega t)$$

$k$ is the wave number given by $$\frac{2\pi}{\lambda}$$

It describes the number of peaks of the wave over a given distance at an instant in time.

$\omega$ is the angular frequency given by $$2\pi\nu$$ where $\nu$ is the frequency, or the reciprocal of the period.

It describes the rate at which you will see peaks travelling through a fixed point in space.

Can you see the whole picture now ?

Last edited: Jan 23, 2005
9. Jan 23, 2005

### Shay10825

The velocity will be multiplied by .25. So why does the graph stay the same??

10. Jan 23, 2005

### christinono

Looking at the equation: v=frequency*wavelength, you can see that the frequency is inversely proportional to the wavelength. Therefore, if you half the frequency, the wavelength is doubled (if the velocity stays constant, of course). Now if you half the wavelength, the frequency doubles. Can you see how these 2 cancel each other out?

11. Jan 23, 2005

### christinono

No, I don't believe it should. If you were simply reducing the wavelenth in half, the graph would look "squished" horizontally by a factor of two. If you were simply reducing the frequency by a factor of 2, the graph would stretch horizontally by a factor of 2 (if all the other variables such as velocity and amplitude were kept constant, of course). But you are doing both, which means the effects cancel each other out.

12. Jan 23, 2005

### Shay10825

oh. I see it. Thanks

13. Jan 23, 2005

### Shay10825

What does a and a_0 stand for?

14. Jan 23, 2005

### christinono

i think a is amplitude.

15. Jan 23, 2005

### Curious3141

a = Displacement (Amplitude) of oscillation at a given point in time and space. It is the vertical coordinate of your graph.

a_0 = Maximum amplitude of oscillation. It is the maximum vertical point your graph reaches.

16. Jan 23, 2005

### learningphysics

The graph should be compressed by a factor of 2. The answer given is wrong. The frequency change does not affect the shape of the graph.

The equation of the wave at a particular time t1 is (modifying Curious' equation):
$$a = a_0\cos[2\pi(\frac{x}{\lambda} - ft_1)]$$

a_0 is just the amplitude of the wave.

if you factor out the lambda, it is:

$$a = a_0\cos[\frac{2\pi}{\lambda}(x - \lambda ft_1)]$$

So when $$\lambda$$ becomes 1/2 of the previous value, the coefficient becomes 2, so the graph compresses by a factor of 2.

If f becomes 1/2 of the previous value then the overall shape is unchanged, only the shift along the x-axis is changed.

Last edited: Jan 23, 2005
17. Jan 23, 2005

### christinono

I don't agree. The wavelength is dependant on the frequency if the speed of the string is kept constant.

18. Jan 23, 2005

### learningphysics

The question didn't mention speed being constant. Besides, if speed is constant, then it is impossible for both frequency and wavelength to be 1/2 of before.

19. Jan 23, 2005

### Shay10825

I have 1 more question.

For a sinusodial wave on a string, if y= 2sin(4x-3t) then a is?

velocity $$v=\frac{dy}{dt}$$
acceleration $$a=\frac{dv}{dt}=\frac{d^2y}{dt^2}$$