Mechanical Waves: Properties and Behaviour Study Notes: Properties and Behaviour Notes on Properties and Behavior

Elastic Materials: A material that returns to its original shape after it has been stretched or compressed is defined as being elastic. This process involves the concepts of displacement and equilibrium.
Inelastic Materials: Materials that do not resume their original shape after being distorted are classified as inelastic.

Restoring Force: Any material or object that possesses a restoring force—an internal force that maintains its original shape—will tend to vibrate if it is disturbed.
Definition of Vibration: A vibration is a repeating back-and-forth motion about an equilibrium position.
Oscillatory Motion: Also known as vibration, this motion is caused by a restoring force. A "Fun Fact" noted is that oscillatory motion is sinusoidal in nature, typically plotted with displacement vs. time, showing peaks at amplitude $A$ and troughs at $-A$ .

Wave Source: The source of all waves is a vibration.
Wave Definition: A wave is a moving disturbance. If a vibrating material is connected to more material, it passes that vibration along. The original disturbance is transmitted (passed along) progressively from one place to the next with no actual transport of matter.
Wave Motion vs. Particle Motion:
- The wave itself moves away from the source.
- The wave medium merely oscillates about some equilibrium point.

Longitudinal Waves: In longitudinal waves, the motion of the particles is parallel to the direction the wave is traveling. These waves consist of regions of compression and expansion (rarefaction). Examples include sound waves and a slinky pushed forwards and backwards.
Transverse Waves: In transverse waves, the motion of the particles is perpendicular (at right angles) to the direction the wave is traveling. These waves consist of crests and troughs. Examples include vibrations in a rope, a vibrating guitar string, and a slinky moved with an upward pulse.
Water Waves: Water waves represent a combination of both transverse and longitudinal waves.

There are fundamentally different kinds of wave mediums depending on the wave type:

Mechanical Waves: These waves require a physical medium to travel through (e.g., sound, water ripples).
Electromagnetic Waves: These do not require a medium. A light wave consists of oscillating electric ( $E$ ) and magnetic ( $B$ ) fields perpendicular to each other and the direction of wave travel.
Gravity Waves: A distinct category of waves involving the curvature of spacetime.

All waves possess the following properties:

Amplitude ( $A$ ): The maximum distance from the midpoint (equilibrium) of the wave. The SI unit is meters ( $m$ ).
Wavelength ( $\lambda$ ): The distance between any two identical points on a wave (e.g., crest to crest). The SI unit is meters ( $m$ ).
Period ( $T$ ): The time required to complete one full cycle. The SI unit is seconds ( $s$ ).
Frequency ( $f$ ): The number of vibrations or waves passing a certain point per second. It is the number of "anything" per unit of time. The SI unit is hertz ( $Hz$ ) or $s^{-1}$ .
Energy: All waves carry energy.
Phase: Particles or waves can be "in phase" (moving together) or "out of phase".

Relationship between Frequency and Period:
- $f = \frac{1}{T}$
- $T = \frac{1}{f}$
Wave Speed ( $v$ ):
- A wave travels a distance of one wavelength ( $\lambda$ ) in the time of one period ( $T$ ).
- Basic formula: $v = \frac{\lambda}{T}$
- Since $f = \frac{1}{T}$ , the wave formula is: $v = f \lambda$
Standard Wave Speeds:
- Speed of sound in air at $20\,^\circ\text{C}$ ( $v_{sound}$ ): $340\,m/s$
- Speed of light ( $v_{light}$ or $c$ ): $3 \times 10^8\,m/s$

7.1 Review Key Questions

1. Particle Motion: Describe particle motion as a mechanical wave passes. (Answer: Particles oscillate about an equilibrium position and do not travel with the wave).
2. True/False:
- a. Longitudinal waves occur when particles vibrate in the opposite direction to the wave. (False: They vibrate parallel to the direction).
- b. Transverse waves occur when vibration is at right angles to the direction. (True).
- c. A longitudinal wave can travel through air. (True).
- d. A vibrating guitar string is a transverse wave. (True).
3. Slinky Pulse: If a student at point A creates a transverse pulse on a slinky, determine the motion of point B. (Point B will move up and then down as the pulse passes).
4. Everyday Examples: Which are mechanical? (Sound, ripples on a pond, and vibrations in a rope are mechanical. Light is electromagnetic).
5. Tuning Fork: How is energy transferred to point X? (Vibrations of the fork compress and rarefy air particles, passing energy through a longitudinal sound wave).
6. Air Particles: Describe how particles A and B form a compression. (They move from undisturbed positions toward each other to create a high-pressure region).
7. Transverse Comparison: How does particle motion compare to wave travel? (It is perpendicular).
8. Classification:
- a. Sound: Longitudinal
- b. Guitar string: Transverse
- c. Slinky upward pulse: Transverse
- d. Slinky forwards/backwards: Longitudinal

7.2 Review Key Questions

1. Graphing (Displacement-Distance):
- a. Two points in phase (e.g., A and E).
- b. Name for distance between these points (Wavelength).
- c. Particles with maximum displacement (B and D).
- d. Term for maximum displacement (Amplitude).
3. Calculation: Displacement-time graph shows a cycle completed in $0.4\,s$ .
- Period $T = 0.4\,s$
- Frequency $f = \frac{1}{0.4} = 2.5\,Hz$
4. Kathy's Wave: 5 wavelengths pass per second ( $f = 5.00\,Hz$ ), wavelength ( $\lambda$ ) is $1.35\,m$ . Speed calculation: $v = f \lambda = 5.00 \times 1.35 = 6.75\,m/s$ .
7. Period Calculation: Frequency is $2.55 \times 10^5\,Hz$ . $T = \frac{1}{2.55 \times 10^5} \approx 3.92 \times 10^{-6}\,s$ .
8. Police Siren: A car moving at the same speed as the police car ( $105\,km\,h^{-1}$ ) would hear the siren at the same frequency because there is no relative motion between the source and the observer.
9. Ambulance: Moving toward you: higher pitch (frequency). Moving away: lower pitch.

Complex Problem: Circular Metal Fence

Scenario: A student hits a circular metal rail. Student 2 stands opposite. Two sounds are heard $0.1\,s$ apart. One travels through air, one through metal.
Given Data:
- $f = 350\,Hz$
- $\Delta t = 0.1\,s$
- $v_{air} = 330\,m/s$
- $v_{metal} = 1310\,m/s$
Geometry:
- Distance in air ( $d_{air}$ ) is the diameter: $2r$
- Distance in metal ( $d_{metal}$ ) is the semi-circle: $\pi r$
Formula:
- $t_{air} - t_{metal} = 0.1\,s$
- $\frac{d_{air}}{v_{air}} - \frac{d_{metal}}{v_{metal}} = 0.1\,s$
- $\frac{2r}{330} - \frac{\pi r}{1310} = 0.1$
Solution: Solving for $r$ yields $r = 79\,m$ .