1. Chapter 15
Mechanical Waves
© 2016 Pearson Education, Inc.

2. Learning Goals for Chapter 15
Understand relationship between speed, frequency, & wavelength for a periodic wave.
Calculate speed of waves on a rope or string.
Identify when mechanical waves overlap & interfere.
Understand properties of standing waves on a string
Grasp how stringed instruments produce sounds of specific frequencies.

3. Introduction Earthquake are waves that carry enormous power
Other types of mechanical waves, e.g. sound waves or the vibration of the strings of a piano, also carry (far less) energy.
Overlapping waves “interfere” which helps us understand musical instruments.

4. Types of mechanical waves
Wave on string is a mechanical wave.
Hand moves string up & down, producing transverse wave.
Wave moves to the right with a speed.

5. Types of mechanical waves
Pressure waves in a fluid are also mechanical waves.
Piston moves right, compressing gas or liquid, & returns, producing a longitudinal wave
Wave moves to the right with a speed.
Particles move right and left

6. Types of mechanical waves
Surface wave on liquids are mechanical waves.
Board moves right & then returns, producing a combination of longitudinal & transverse waves.

7. Mechanical waves
“Doing the wave” at a sports stadium is a mechanical wave.
Wave disturbance propagates through crowd, but there is no transport of matter.
None of the spectators moves from one seat to another.

8. Periodic waves
For periodic waves, each particle of medium undergoes periodic motion.
Wavelength λ (“lambda”) of periodic wave is length of one complete wave pattern (units: Distance!)
Speed of periodic wave of frequency f is:

9. Periodic transverse waves
Mass attached to spring undergoes SHM, producing sinusoidal wave traveling right on string:
Wavelenth l

10. Periodic transverse waves
Mass attached to spring undergoes SHM, producing sinusoidal wave traveling right on string:

11. Periodic waves
Series of drops falling into water produces a periodic wave that spreads radially outward.
Wave crests & troughs are concentric circles.
Wavelength λ is radial distance between adjacent crests or adjacent troughs.

12. Periodic longitudinal waves
Consider long tube filled with fluid, with piston at left end.
Push piston in, compress fluid near piston
This region pushes against neighboring region of fluid,
And so on… and a wave pulse moves along the tube.

13. Periodic longitudinal waves

14. Mathematical description of a wave
Wave function for sinusoidal wave moving in +x-direction:
y = displacement of particle at time t & position x (distance)
A = amplitude of wave (distance)
k = wave number, defined as k = 2π/λ (m-1)
ω = angular frequency, defined as ω = 2πf = 2π/T (rad/sec)

15. Graphing the wave function
y(x, t) =𝐴𝑐𝑜𝑠 𝑘𝑥−𝑤𝑡
𝑦 𝑥 = 𝐴𝑐𝑜𝑠 𝑡=0

16. Graphing the wave function
y(x, t) at a particular time t is a
SNAPSHOT in time
graph of the entire wave at that instant
𝑦 𝑥 = 𝐴𝑐𝑜𝑠 𝑡=0

17. Graphing the wave function
y(x, t) =𝐴𝑐𝑜𝑠 𝑘𝑥−𝑤𝑡
𝐴𝑐𝑜𝑠 𝑥=0

18. Graphing the wave function
y(x, t) =𝐴𝑐𝑜𝑠 𝑘𝑥−𝑤𝑡
y(x, t) at a particular position x is a
MOVIE graph of the wave at that point only
𝐴𝑐𝑜𝑠 𝑥=0

19. Particle velocity and acceleration in a sinusoidal wave

20. Particle velocity and acceleration in a sinusoidal wave

21. The speed of a wave on a string
Key property of any wave is wave speed.
Consider string with tension F & linear mass density (mass per unit length) is μ.
Expect speed of transverse waves on string v to increase when tension F increases
Expect wave speed to decrease when μ increases.
Wave speed is:

22. The speed of a wave on a string
Transmission cables w/ relatively large mass per unit length, & low tension.
If cables disturbed—say, by birds —transverse waves will travel at a slow speed.

23. Power in a wave Instantaneous power in a sinusoidal wave
Power is never negative!
Energy never flows opposite to the direction of wave propagation.

24. Power in a wave Waves transfer power because they transfers energy.
Average power proportional to square of amplitude
Average power proportional to square of frequency.
This result is true for all waves.
For transverse wave on string, average power is:

25. Wave intensity
Intensity of a wave is average power it carries per unit area.
Units: Watts/m2
IF waves spread out uniformly in all directions & NO energy is absorbed
I at any distance r from a wave source is inversely proportional to r2.

26. Reflections of wave pulses @ fixed end
What happens when a wave pulse or a sinusoidal wave arrives at the end of the string?
If end is fastened to a rigid support, it is a fixed end that cannot move.
Arriving wave exerts force on fixed support….

27. Reflection of a wave pulse at a fixed end of a string
Reaction force exerted by support on string “kicks back” on string
Reaction force sets up a reflected pulse or inverted wave traveling in reverse direction.

28. Reflection of wave pulse at free end
Free end is one that is perfectly free to move in direction perpendicular to length of string.
When wave arrives at free end, ring slides along rod, reaching maximum displacement, coming momentarily to rest…

29. Reflection of a wave pulse at a free end of a string
String is now stretched, giving increased tension,
Free end is pulled back down
A reflected pulse is produced
But NOT inverted!

30. Superposition
Interference is result of overlapping waves at a point in time.
Principle of superposition: When two or more waves overlap, total displacement is sum of displacements of individual waves.

31. Superposition
Overlap of two wave pulses— one right side up, one inverted—traveling in opposite directions.
Time increases from top to bottom.

32. Superposition
Overlap of two wave pulses— one right side up, one inverted—traveling in opposite directions.
Time increases from top to bottom.

33. Standing waves on a string
Waves traveling in opposite directions on a taut string interfere with each other.
Result is standing wave pattern that does not move on the string.
Destructive interference occurs where the wave displacements cancel, and constructive interference occurs where the displacements add.
At the nodes no motion occurs, and at the antinodes the amplitude of the motion is greatest.

34. Standing waves on a string
Waves traveling in opposite directions on a taut string interfere with each other.
Result is standing wave pattern that does not move on the string.

35. Standing waves on a string
This pattern is called the second harmonic.

36. Standing waves on a string
As frequency of oscillation of right-hand end increases, pattern of standing wave changes.
More nodes & antinodes are present in higher frequency standing wave.

37. The mathematics of standing waves
Derive wave function for standing wave by adding two wave functions for waves w/ equal amplitude, period, & wavelength traveling in opposite directions.
Wave function for standing wave on string where x = 0 is a fixed end is:
Standing-wave amplitude ASW is twice the amplitude A of either of the original traveling waves: ASW = 2A.

38. Normal modes f1 is the fundamental frequency Wavelength l1 = 2 L
Wave speed v = f1 l1

39. Normal modes f1 is the fundamental frequency
f2 is the second harmonic (first overtone)
Wavelength l2 = L
Wave speed v = f2 l2
Wave speed v hasn’t changed!
Same T, same m!

40. Normal modes f1 is the fundamental frequency,
f2 is the second harmonic (first overtone),
f3 is the third harmonic (second overtone), etc.

41. Normal modes
For a taut string fixed at both ends, possible wavelength:
Possible frequencies are fn = n v/2L = nf1
where n = 1, 2, 3, …

42. Standing waves and string instruments
Move string on a musical instrument (pluck, bow, or strike), a standing wave with fundamental frequency is produced:
This is also frequency of sound wave created in surrounding air by vibrating string.
Increasing tension F increases frequency (and pitch).