# Waves and vibrations on a string

• Diku Khanikar
In summary, the conversation discusses the calculation of various parameters related to a stretched string fixed at both ends with a given length and mass per unit length. These include the wavelength of the first two harmonics, the speed of the wave, and the frequency of the third and fourth harmonics. The conversation also delves into the potential energy density of a vibrating string, represented by the equation E=(1/2)T(∂y∂x)^2, where T is the tension in the string and y=y(x,t) is the transverse displacement of the string. The solution to the last question is provided through a link to a document.
Diku Khanikar
Homework Statement
Relevant Equations
E=(1/2)T(∂y∂x)^2
Q.1. The length of a stretched string fixed at both ends has a length of l=10 cm, mass per unit length ρ= 0.01 gm/cm. If the tension ' T ' is produced by hanging a 11 kg weight at both ends of the string, then calculate,

a) The wavelength of the first two harmonics,

b) The speed of the wave

c) The frequency of the third and fourth harmonics

Q. 2. Prove that the potential energy density of a vibrating string is given by
E=(1/2)T(∂y∂x)^2
where T is the tension in the string and y=y(x,t) is the transverse displacement of the string.

That question took a turn! The first three should follow from consideration of ##v_p = f\lambda##, whilst the last is a bit more difficult.

It's hard to give a hint without giving the whole thing away (and the derivation I know is pretty hand-wavy anyway); maybe try considering the EPE of small 'arc' of string?

## 1. What are waves and vibrations on a string?

Waves and vibrations on a string refer to the movement and energy transfer of a string as it oscillates back and forth. These vibrations create a disturbance in the string, causing a wave-like pattern to propagate along its length.

## 2. How are waves and vibrations on a string measured?

Waves and vibrations on a string can be measured using various techniques such as frequency analysis, amplitude measurement, and phase measurement. These measurements can provide information about the characteristics of the wave, such as its frequency, wavelength, and speed.

## 3. What factors affect the behavior of waves and vibrations on a string?

The behavior of waves and vibrations on a string is affected by several factors, including tension, length, and density of the string, as well as the properties of the medium through which the wave is traveling. These factors can impact the frequency, wavelength, and speed of the wave.

## 4. How do waves and vibrations on a string relate to music?

Waves and vibrations on a string play a crucial role in the production of music. When a string is plucked or struck, it creates vibrations that travel through the instrument and produce sound waves. The frequency and amplitude of these waves determine the pitch and volume of the sound produced.

## 5. What are some real-life applications of waves and vibrations on a string?

Waves and vibrations on a string have various applications in everyday life, such as in musical instruments, telecommunications, and medical imaging. They also play a role in fields like seismology, where they are used to study and understand earthquakes and other seismic activity.

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