Question

The equation y(x,t)=Acos2πf(xv−t) may be written as y(x,t)=Acos[2πλ(x−vt)].

Use the last expression for y(x,t) to find an expression for the transverse velocity vy of a particle in the string on which the wave travels. Express your answer in terms of the variables A, v, λ, x, t, and appropriate constants.

Find the maximum speed of a particle of the string. Express your answer in terms of the variables A, v, λ, x, t, and appropriate constants.

Answer 1

Concepts and reason

The concepts used to solve this problem are the transverse wave in a string, transverse velocity of a point in the wave and the maximum velocity of the particle travelling along the wave.

First find the general equation of the transverse wave.

Find the transverse velocity of the particle in the string on which the wave is found by differentiating the wave equation.

Finally find the maximum speed of the particle in the string using the extreme value of sine.

Fundamentals

The general expression for the transverse wave is,

Here, is the wave, is the amplitude of the wave, is the wavelength of the wave, is the frequency of the wave, is the position of the wave, and is the period of the wave.

The velocity of a particle of the string on which the wave travels is expressed as,

Here, is the transverse velocity of a particle in the wave.

The maximum velocity is expressed as,

Here, is the maximum transverse velocity, is the amplitude, is the wavelength, and is the frequency of the wave.

(1)

The general equation of the transverse wave travelling in a string is,

The velocity of a particle in the travelling wave is,

Substitute for and differentiate to find the expression for the transverse velocity.

(2)

The value of sine is bound between.

The maximum value of the velocity, for a given frequency and wavelength, corresponds to the term taking the value +1.

Therefore,

Ans: Part 1

The transverse velocity of a point in the string on which the wave travels is .

Part 2

The maximum transverse velocity of the particle in the string is .