Standing waves on a fixed string

[BOAS]

Hello,

Homework Statement

Two wires, each of length 1.8 m, are stretched between two fixed supports. On wire
A there is a second-harmonic standing wave whose frequency is 645 Hz. However,
the same frequency of 645 Hz is the third harmonic on wire B. Find the speed at
which the individual waves travel on each wire.

Homework Equations

$L = \frac{nv}{2f_{n}}$

The Attempt at a Solution

I don't know if I understand the idea of natural frequencies correctly and it's relation to n (an integer value in the above equation).

If I imagine a string fixed at both ends there are a number of different standing waves that can be made, ie different harmonics.

The first harmonic has 1 antinode, the second has two etc.

When working out the velocity of the wave on a string, does the 'n' refer to the harmonic? I assume that the different harmonics can be considered to be the natural frequencies of the string.

I'm fairly sure I have this wrong, because I get slower speeds for higher harmonics and intuition tells me that this is wrong. I remember having to shake the rope up and down much harder to reach the next standing wave in an 'experiment' that was done in school.

I'd really appreciate a helping hand :)

BOAS

 

[nasu]

The speed is the same for all harmonics (and fundamental). At least in the first approximation.
So in your formula, for a given string, L and v are fixed. Only frequency and n change for various harmonics.

 

[AlephZero]

I'm fairly sure I have this wrong, because I get slower speeds for higher harmonics and intuition tells me that this is wrong. I remember having to shake the rope up and down much harder to reach the next standing wave in an 'experiment' that was done in school.

In your experiment, the rope was the same length and tension, and the frequency changed.

In this question, you have two different wires, with different tensions and/or diameters, and the frequency stays the same.

Your "relevant equation" is correct.

 

[BOAS]

In your experiment, the rope was the same length and tension, and the frequency changed.

In this question, you have two different wires, with different tensions and/or diameters, and the frequency stays the same.

Your "relevant equation" is correct.

So rearranging for v gives $v = \frac{2fl}{n}$ and the n refers to the particular harmonic, hence the standing wave on the second string is slower.

Thanks for the help - it hadn't twigged in my brain that the two strings are not said to be the same.

BOAS

 

[HalfLostAndSearching]

I can provide you with a response to the content you have presented.

Standing waves on a fixed string are a fundamental concept in physics and can be observed in many different systems, from musical instruments to power lines. In this scenario, we have two wires of equal length and different natural frequencies, which are determined by the fixed supports and the tension of the wire. The natural frequency of a string is directly related to its length and the speed at which waves travel on the string.

In this case, we are given that wire A has a second-harmonic standing wave with a frequency of 645 Hz, while wire B has a third-harmonic standing wave with the same frequency. This means that wire A has a shorter wavelength compared to wire B, as the second harmonic has one more antinode than the third harmonic. Using the equation L = (n/2) * v/f, where L is the length of the string, n is the harmonic number, v is the velocity of the wave, and f is the frequency, we can solve for the velocity of the wave on each wire.

For wire A, we have L = 1.8 m, n = 2, and f = 645 Hz. Plugging these values into the equation, we get v = 2 * 645 Hz * 1.8 m / 2 = 1161 m/s.

For wire B, we have L = 1.8 m, n = 3, and f = 645 Hz. Plugging these values into the equation, we get v = 3 * 645 Hz * 1.8 m / 3 = 1161 m/s.

As you can see, the velocity of the wave on both wires is the same, despite the different harmonics. This is because the natural frequency of a string is determined by its physical properties and does not depend on the harmonic number. Therefore, the speed at which the individual waves travel on each wire is 1161 m/s.

I hope this explanation helps you understand the concept of natural frequencies and how they relate to the velocity of waves on a string. If you have any further questions, please don't hesitate to ask.