## Friday, February 8, 2008

### IIT JEE Physics Formula Revision 15. Wave motion and waves on a string

1. Equation of a wave travelling in the positive x-direction with a constant speed v.

The displacement of the particle at x at time t i.e., y(x,t) is generally abbreviated as y and the wave equation is written as

y = f(t - x/v) ... (1)

2. Equation of a wave travelling in the negative x-direction with a constant speed v.

The displacement of the particle at x at time t i.e., y(x,t) is generally abbreviated as y and the wave equation is written as

y = f(t + x/v) ... (2)

3. The wave equation in (1) can be written as y = f((vt-x)/v) which can be transformed into y = g(x-vt)...(3)

g is a different function. Function g can have the following meaning. If you put t = 0 in equation (3), you get the displacement of various particles at t = 0;

y(x, t = 0) = g(x)

If displacement at t = 0 of all particles of the string is represented by g(x) then the displacement of the particle at x at time t will be y = g(x-vt).

4. Similarly if the wave is travelling along the negative x-direction and the displacement of different particles at t = 0 is g(x), the displacement of the aprticle at x at time t will be

y = g(x+vt)

Function f in equation 1 and 2 represents the displacement of the point x = 0 as time passes,and g in (3) and (4) represents the diplacement at t = 0 of different particles.

5. sine wave or sinusoidal wave

When a person vibrates the left end of a string x = 0 in a simple harmonic motion, the equation of motion of this end may be written as

f(t) = A sin ωt ... (5)

A represents the amplitude
ω = the angular frequency
Time period of oscillation is T = 2π/ω
Frequency of oscillation = 1/T = ω/2π

This wave is called a sine wave or sinusoidal wave.

If the displacement of the particle at x = 0 is given by f(t) = A sin ωt, the displacement of the particle at x at time t will be given by

y = f(t - x/v) = A sin ω(t - x/v)...(6)

7. Velocity of the particle at x at time t is given by

ðy/ðt = Aω cos ω(t - x/v)...(7)

This velocity is different from the velocity of the wave. The wave moves on the string at a constant velocity v along the x axis, but the particles at various points move up and down with velocity ðy/ðt which changes with x and t according to equation (7).

8. λ = (v/ω)2π = vT .... (8)

9. v = λ/T = νλ ... (9)

where ν = 1/T is the frequency of the wave.

Alternative sine wave equations

y = A sin ω(t - x/v)

y = A sin (ωt - kx)... (10)

y = A sin 2π(t/T - x/λ)... (11)

y = A sin k(vt-x) ... (12)

General equation will be

y = A sin [ω(t - x/v)+ Φ)... (13)

Φ is the phase constant.

This equation will allow us to write equation based on the displacement of the left at t = 0.

The constant Φ will be π/2 is we choose t = 0at an instant when the left end reaches its extreme position y = A. The equation will then be

y = A cos ω(t - x/v)... (14)

If t = 0 is taken at the instant when the left end is crossing the mean position from upward to downward direction, Φ will be π and the equation will be

y = A sin (kx-ωt)... (15)

16. Velocity of a wave on a string

v = √(F/μ)

F = tension in the string
μ = linear mass density of the string ... (16)

17. Power transmitted along the string by a sine wave

Pav = 1/2 [ω² A² F²/v]... (17)