# Overtones in a string (equation for wave)

• Incand
In summary, the conversation discusses determining the wave equation for the first harmonic of a string with a fundamental tone of ##s_1 = A_1\sin (\omega_1 t - k_1 x)##, where the sound intensity level of the harmonic is 20dB lower than the fundamental tone. The equations for sound intensity level ##L = 10\lg (\frac{I_1}{I_0} )## and sound intensity ##I \sim A^2## are used to find the amplitude relationship ##\frac{A_2}{A_1} = 0.1## for the first harmonic with double the frequency ##\omega_1 = 1360/s##. However, the correct
Incand

## Homework Statement

A string has the fundamental tone of
##s_1 = A_1\sin (\omega_1 t - k_1 x)##
Determinate the wave equation for the first harmonic of the string if the sound intensity level of the harmonic is 20dB lower than the fundamental tone. ##\omega_1 = 1360/s## and ##k_1 = 4/m##.

## Homework Equations

Sound intensity level
##L = 10\lg (\frac{I_1}{I_0} )##
Sound intensity is proportional to the wave amplitude squared

##I \sim A^2##

## The Attempt at a Solution

Sound intensity level
##-20 = 10\lg( \frac{I_2}{I_1}) \Longleftrightarrow I_2 = 0.01I_1##
Amplitude relationship
##\frac{A_2}{A_1} = 0.1 \Longleftrightarrow A_2 = 0.1A_1##
The first harmonic got double the frequency so
##s_2 = 0.1A_1 \sin(2 \omega_1 t - 2 k_1 x)##
which is wrong. According to the answer key it should be
##s_2 = 0.05A_1\sin(2 \omega_1 t-2k_1 x)##
Why is it 0.05?

As I recall the intensity of a wave is proportional to the square of its amplitude. But perhaps I'm rusty.

TSny said:
Energy of a wave on string depends on ##\omega## as well as ##A##.

See http://hyperphysics.phy-astr.gsu.edu/hbase/waves/powstr.html
Cheers!

CWatters said:
As I recall the intensity of a wave is proportional to the square of its amplitude. But perhaps I'm rusty.
Always a bit tricky, luckily we got TSny to link us the formula :)

## 1. What are overtones in a string?

Overtones in a string refer to the higher frequency vibrations that occur in addition to the fundamental frequency when a string is plucked or struck.

## 2. How are overtones related to the length of a string?

The length of a string determines the wavelengths of the overtones that can be produced. As the string length decreases, the wavelengths of the overtones also decrease, resulting in higher frequency vibrations.

## 3. What is the equation for calculating the frequency of an overtone in a string?

The equation for calculating the frequency of an overtone in a string is: f = nv/2L, where n is the overtone number, v is the velocity of the wave, and L is the length of the string.

## 4. How do overtones affect the sound produced by a string instrument?

Overtones add complexity and richness to the sound produced by a string instrument. They contribute to the overall timbre and can be manipulated by the musician to create different tones and effects.

## 5. Can overtones be heard in all types of strings?

Yes, overtones can be heard in all types of strings, including guitar strings, violin strings, and even vocal cords. They are a natural result of the vibrating string and contribute to the unique sound of each instrument.

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