Harmonic Wave Train

Harmonic Wave Train

If the source of the waves is a simple harmonic oscillator, the function f(x ± vt) is sinusoidal and it represents a harmonic wave train or simply, a travelling harmonic wave. When such a wave passes through a given region, the particles of the medium execute simple harmonic motion.

The wave function that represents a harmonic wave is given by

y = A sin [k(x ± vt)]

or y = A sin (kx ± ωt) (14.5)

The negative sign is used when the wave travels along the positive x – axis, and vice-versa.

The term k is called the wave number which is defined as

k = 2π/λ (14.6)

where λ is called the wavelength. It is the distance between the two successive points with the same phase (for example, two crests).

The term ω is called the angular frequency (measured in rad/s)

ω = 2π/T = 2πf (14.7)

where T is the time period and f is the frequency.

Time Period (T)

The time period for a point on the string is the time taken to complete one cycle of its periodic motion. It is exactly the same time that it takes for one wavelength to pass the point.

Frequency ( f )

The number of complete vibrations of a point on the string that occur in one second or, the number of wavelengths that pass a given point in one second.

Wave velocity (v)

Since in one period T the wave advances by one wavelength λ, therefore, the wave velocity is

v =2π/λ    (14.8)

Amplitude (A)

The maximum displacement of a particle on the medium from the equilibrium position

IMPORTANT

1. The photograph of a travelling harmonic wave at a given instant (as shown in Fig.14.5 a) shows that the displacement of various points on the string follows a sine curve.

In equation (14.5) if the time t is kept constant at t = 0, it reduces to

y = A sin kx(14.9)

2. If we focus our attention on a particular particle of the string, we find that its displacement also follows a sinusoidal function with respect to time as shown in the Fig.(14.5 b).

In Fig.(14.5 b) if the particle is at the origin

y = -A sin ωt (14.10)

General Equation of a Harmonic Wave

y = A sin (kx ± ωt + φ)     (14.11)

where φ is the initial phase constant. It determines the initial displacement of the particle when
x = 0 and t = 0. The positive and negative value of φ are shown in (Fig.14.6).

Wave Velocity and Particle Velocity

Wave velocity is the velocity of the disturbance which propagates through a medium. It only depends on the properties of the medium and is independent of time and position.

Particle velocity is the rate at which particle’s displacement vary as a function of time, i.e.

δy/δt = ± ωAcos (kx ± ωt + φ)    (14.12)

The difference between the wave velocity v and the velocity of a particle of the medium (δy/δt) is shown in the Fig.(14.7).

The acceleration of the particle is obtained by differentiating equation (14.12) w.r.t. time

δ²y/δt² = ± ω²Asin (kx ± ωt + φ)   (14.13)

Using equation (14.11) in equation (14.13), we have

δ²y/δt²  = - ω²y (14.14)

The above equation shows that acceleration of a particle is directly proportional to its displacement from the equilibrium position, i.e. the particles of the medium execute simple harmonic motion.

Example 14.2

The equation of a transverse wave in a stretched wire is given by, y = 2sin2π(x/30 - t-0.01) where t is in s and y in cm.

Find :

(a) amplitude

(b) frequency

(c) wavelength and

(d) speed of the wave

Solution

Comparing the given equation with the standard equation (14.11),

y = A sin(2π/λx - 2π/Tt), we obtain

(a) Amplitude, A = 2 cm.

(b) Time period, T = 0.01s.

∴ Frequency, f = 1/T = 100 Hz

(c) Wavelength λ = 30 cm

(d) Velocity of the wave v = λ f = 30 × 100 = 3000 cm/s = 3 m/s