1. Chapter 16
Waves (I)
What determines the tones of strings on a guitar?

2. Key contents
Types of Waves
Wave Variables
The Speed of a Traveling Wave
Energy and Power of a Wave Traveling along a String
The Wave Equation
The Superposition of Waves
Interference of Waves
Standing Waves and Resonance

3. Electromagnetic waves
16.1 Types of Waves
Mechanical waves
Electromagnetic waves
Gravitational waves
Matter waves
# Waves are propagating ‘disturbance’ without matter transportation, but energy and momentum are transported.

4. In a longitudinal wave the motion of
16.1 Types of Waves
In a transverse wave, the displacement of every such oscillating element along the wave is perpendicular to the direction of travel of the wave, as indicated in Fig
# EM waves in vacuum are also called transverse waves, because the direction of fields is perpendicular to the propagation direction.
In a longitudinal wave the motion of
the oscillating particles is parallel to the direction of the wave’s travel, as shown in Fig
# In this chapter, we will focus on transverse mechanical waves.

5. 16.2 Wave Variables

6. 16.2 Wave Variables
The amplitude ym of a wave is the magnitude of the maximum displacement of the elements from their equilibrium positions as the
wave passes through them.
The phase of the wave is the argument (kx –wt) of the sine function. As the wave sweeps through a string element at a particular position x, the phase changes linearly with time t.
The wavelength l of a wave is the distance parallel to the direction of the wave’s
travel between repetitions of the shape of the wave (or wave shape). It is related to the angular wave number, k, by :
The period of oscillation T of a wave is the time for an element to move through one full oscillation. It is related to the angular frequency, w, by
The frequency f of a wave is defined as 1/T and is related to the angular frequency w by
A phase constant f in the wave function:
y =ym sin(kx –wt+ f). The value of f can be chosen so that the function gives some other displacement and slope at x = 0 when t = 0.

7. 16.3 The Speed of a Traveling Wave

8. Example, Transverse Wave

9. Example, Transverse Wave, Transverse Velocity, and Acceleration

10. Dimension analysis : what factors should come in?
16.4 Wave Speed on a Stretched String
Dimension analysis : what factors should come in?
Tension [t] = ML/T2
Density [r] = M/L3
Area of cross section [A] = L2
 Restoring force factor
 Inertia factor

11. 16.4 Wave Speed on a Stretched String
The speed of a wave along a stretched ideal string depends only on the tension and linear density of the string and not on the frequency of the wave.
A small string element of length Dl within the pulse is an arc of a circle of radius R and subtending an angle 2q at the center of that circle. A force with a magnitude equal to the tension in the string, t, pulls tangentially on this element at each end. The horizontal components of these forces cancel,
but the vertical components add to form a radial restoring force . For small angles,
If m is the linear mass density of the string, and Dm the mass of the small element,
The element has an acceleration:
Therefore,

12. 16.5 Energy and Power of a Wave Traveling along a String
The average power, which is the average rate at which energy of both kinds (kinetic energy and elastic potential energy) is transmitted by the wave, is:

13. Example, Transverse Wave:

14. A travelling wave is always in the following form:
16.6 The Wave Equation
A travelling wave is always in the following form:
Such functions are solutions of the wave equation:
# It is a linear partial differential equation; when y1 and y2 are solutions, any linear combination of y1 and y2 (like ay1+by2) is also a solution.

15. Overlapping waves do not in any way alter the travel of each other.
16.7 The Superposition of Waves
Overlapping waves algebraically add to produce a resultant wave (or net wave).
Overlapping waves do not in any way alter the travel of each other.
They interfere but do not interact.
# The principle of linear superposition is valid only when the amplitude is small.

16. 16.8 Interference of Waves
If two sinusoidal waves of the same amplitude and wavelength travel in the same direction along a stretched string, they interfere to produce a resultant sinusoidal wave traveling in that direction.

17. 16.8 Interference of Waves

18. 16.8 Interference of Waves

19. Example, Transverse Wave:

20. 16.9: Standing Waves

21. 16.9: Standing Waves
If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a stretched string, their interference with each other produces a standing wave.

22. The amplitude is zero when kx =np, for n =0,1,2, . . . .
16.9: Standing Waves
The amplitude is zero when kx =np, for n =0,1,2,
Since k =2p/l, we get x = n l/2, for n =0,1,2, (nodes),
as the positions of zero amplitude or the nodes.
The amplitude has a maximum value of 2ym when
kx = 1/2p, 3/2p, 5/2p, . . .=(n+1/2) p, for n =0,1,2,
That is, x = (n+1/2) l/2, for n 0,1,2, (antinodes),
as the positions of maximum amplitude or the antinodes.

23. * Reflection at a boundary
16.10: Standing Waves and Resonance
* Reflection at a boundary

24. 16.10: Standing Waves and Resonance
For certain frequencies, the interference produces a standing wave pattern (or oscillation mode) with nodes and large antinodes like those in Fig
Such a standing wave is said to be produced at resonance, and the string is said to resonate at these certain frequencies, called resonant frequencies.
Fig Stroboscopic photographs reveal (imperfect) standing wave patterns on a string being made to oscillate by an oscillator at the left end. The patterns occur at certain frequencies of oscillation. (Richard Megna/Fundamental Photographs)

25. 16.10: Standing Waves and Resonance
The frequencies associated with these modes are often labeled f1, f2, f3, and so on. The collection of all possible oscillation modes is called the harmonic series, and n is called the harmonic number of the nth harmonic.

26. Example, Standing Waves and Harmonics:
Figure shows a pattern of resonant oscillation of a string of mass m =2.500 g and length L =0.800 m and that is under tension t =325.0 N. What is the wavelength l of the transverse waves producing the standing-wave pattern, and what is the harmonic number n? What is the frequency f of the transverse waves and of the oscillations of the moving string elements? What is the maximum magnitude of the transverse velocity um of the element oscillating at coordinate x =0.180 m ? At what point during the element’s oscillation is the transverse velocity maximum?
Calculations:
By counting the number of loops (or half-wavelengths) in
Fig , we see that the harmonic number is n=4.
Also,
For the transverse velocity,
We need:
But ym =2.00 mm, k =2p/l =2p/(0.400 m), and w= 2pf =2p (806.2 Hz).
Then the maximum speed of the element at x =0.180 m is

27. Homework:
Problems 13, 23, 27, 50, 58