General Equation of Waves

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11.2General Equation of Waves


In these revision notes for General Equation of Waves, we cover the following key points:

  • What is wave function?
  • What is the form of the general equation of waves?
  • What is angular wave number and how it relates with the wavelength?
  • How to obtain the simplified wave equation from the general equation of waves?
  • How to calculate the speed and acceleration of oscillating particles in a wave?
  • How to interpret wave graphs?

General Equation of Waves Revision Notes

Waves motion is periodical. This means it has many similarities with both circular motion and SHM. The graph of waves motion is sinusoidal, just like in SHM.

Since we are dealing with travelling waves, i.e. with waves that are displaced from the original position, when studying their behavior, we need to consider three quantities instead of two: the x and y coordinates versus time t. Therefore, the equation of waves must be of the type

y(x,t)

Hence, given that the wave equation has a sinusoidal form, we write its general form as

y(x,t) = ymax × sin(k × x - ω × t)

where

ymax represents the amplitude A of a point of the wave, as it oscillates according the y-axis,

ω represents the angular frequency (ω = 2π / T as usual),

x is the horizontal position,

t is the time of motion, and

y(x, t) is the vertical position at a given instant.

The quantity k is known as the angular wave number. It is calculated by

k = /λ

Also, since ω = /T, we can also insert the period T in the wave equation. Therefore, it becomes

y(x,t) = ymax × sin(/λ × x - /T × t)

The argument k × x - ω × t is known as the phase.

If there is a phase shift φ from the original position of wave, we insert its value in the equation of waves. Thus, it becomes

y(x,t) = ymax × sin(k × x - ω × t + φ)

Since the argument is constant,

k × x - ω × t = constant

we obtain from its first derivation with time

k × dx/dt - ω=0
k × v - ω = 0
k × v = ω
v = ω/k

and after substituting the known values,

v = λ × f

The equation

y(x,t) = ymax × sin(k × x - ω × t + φ)

is an equation of particles oscillation, while

v = λ × f

is an equation of waves that carry energy, not particles.

The speed of oscillating particles in a wave is calculated by taking the first derivative of y-position with respect to the time, i.e.

vy (t) = dy/dt
= d[ymax × sin(k × x - ω × t + φ) ]/dt
= ymax × ω × cos(k × x - ω × t + φ)

Likewise, we obtain for the vertical acceleration of oscillating particles,

ay (t) = dv/dt
= -d[ymax × ω × cos(k × x - ω × t + φ) ]/dt
= -ymax × ω2 × sin(k × x - ω × t + φ)
= -ω2 × y(t)

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