14/06/2020
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
6. 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 ve