Summary of Chapter 17
1. Wave is the motion of a disturbance.
2. There are two (2) types of waves:
Transverse Waves: The particles of the disturbed medium move perpendicular to the wave velocity
(example: waves on a string).
Longitudinal Waves: The particles of the disturbed medium move parallel to the wave velocity
(example: sound waves).
3. For a sinusoidal wave the transverse displacement y at position x at time t is given by
This equation represents the displacement of the particles of the medium as a function of x and t.
where ym is the amplitude of the wave in (m), k is the wave number in (rad/m), w is the angular frequency in (rad/s).
The expression () is called the phase angle and its units is radians.
The plus (+) sign means the wave is traveling to the left.
The minus (-) sign means the wave is traveling to the right.
A wave is said to be stationary if is time independent, i.e., y (x) = f (x).
A travelling wave can be longitudinal or transverse.
Ø The transverse velovity (the velocity of the particles) is given by ;
The maximum transverse velocity is umax = ym w (m/s)
Ø The transverse acceleration (the acceleration of the particles) is given by;
The maximum transverse acceleration is amax = ym w2 (m/s2)
4. The speed of the wave is constant and given by
The wavelength l is the distance between two successive maxima (crests) or minima (throughs) or any two identical points on the wave.
The frequency of the wave is also related to the period T in (s) by
5. In the case of a stretched string as in the figure, the wave speed is given by:
where T is the tension in the string in (N), m is the linear mass density (kg/m) and v is the speed of the wave (m/s).
6. The power transmitted by a harmonic wave moving in a stretched string is given by; with units of Watt
7. Superposition of Waves
Suppose two waves y1 (x,t) and y2(x,t) are moving through a medium, the total displacement of the medium y(x,t) at a point x at time t is
y(x,t) is called the resultant wave.
8. Interference of Waves
The net displacement of two equal sinusoidal waves (the resultant) traveling in the same direction equals the algebraic sum of the displacement of the two waves.
y1 = ym sin(k x - w t)
y2 = ym sin(k x - w t + f)
The resultant wave function is y = y1 + y2 = [2 ym cos f/2] sin(k x - w t + f/2)
Amplitude of the resultant
Amplitude of the resultant wave
a) The amplitude of the resultant wave is maximum (constructive interference) when f = 0, 2p, 4p, 6p,
b) The amplitude of the resultant wave is zero (destructive interference) when f = p, 3p, 5p,
c) Intermediate interference when 0 < f < p etc, ..
9. Standing Waves
If two equal sinusoidal waves travel in opposite directions along a stretched string, their interference with each other produce a standing wave.
y1 = ym sin(k x - w t) and y2 = ym sin(k x + w t)
y = y1 + y2 =[ 2 ym sin(k x)] cos(w t)
a) position of the nodes or minimum amplitude:
kx = np, for n = 0, 1, 2, 3,
but k = 2p/l
Þ x = l/2, l, 3l/2, = nl/2 for n = 0, 1, 2, 3,
b) position of the anti-nodes or maximum amplitude:
k x = p/2, 3p/2, 5p/2, = np/2 for n = 0, 1, 2, 3,
but k = 2p/l
Þ x = l/4, 3l/4, 5l/4, = nl/4 for n = 1, 3, 5
A standing Wave
10. Standing Waves and Resonance in a stretched string
In the case of a stretched string resonance occur and standing wave patterns are formed at certain freqencies called resonant frequencies.
The standing wave patterns in the case of a stretched string are as follows:
or first harmonic
Fundamental mode or first harmonic
The relation between the wavelength (l) and the length of the string (L) is:
or for n = 1, 2, 3,
As you can see from the figures, n represents the number of loops or segments.
for n = 1, 2, 3,
In this case we can either fix the tension and varie the frequency to obtain the various harmonics or fix the frequency
and change the tension to obtain the various harmonics.
This latter case was done in the Phys. 102 lab. (Standing waves in a string).