Mechanical Waves: Types, Properties & Equations

What Are Mechanical Waves?

Mechanical waves are disturbances that travel through a medium (solid, liquid, or gas) by transferring energy, not matter. They require a medium to propagate. Examples include:

• Sound waves (travel through air, water, or solids).
• Waves on a string (like a guitar string).
• Water ripples (created by dropping a stone).

Key Features of Mechanical Waves

graph TD
    A["Mechanical Waves"] --> B["Need a Medium"]
    A --> C["Transfer Energy, Not Matter"]
    A --> D["Types: Transverse & Longitudinal"]
    A --> E["Described by: Wavelength, Frequency, Amplitude, Speed"]

Visual:

![transverse and longitudinal wave diagram](/media/3caacd130ab7e3311790.png "Shows how particles move in each type. (Image: Amir Menad, Public domain, via Wikimedia Commons)")

Types of Mechanical Waves

1. Transverse Waves

• Particles move perpendicular to the wave direction.
• Example: Waves on a string, light waves (though light is an electromagnetic wave, not mechanical).
• Crest: Highest point.
• Trough: Lowest point.

2. Longitudinal Waves

• Particles move parallel to the wave direction.
• Example: Sound waves, slinky waves.
• Compression: Region where particles are close.
• Rarefaction: Region where particles are far apart.

Comparison Table:

Wave Properties

1. Wavelength (λ)

• Distance between two consecutive crests or troughs (for transverse waves).
• Distance between two compressions or rarefactions (for longitudinal waves).
• Unit: Meter (m).

2. Frequency (f)

• Number of waves passing a point per second.
• Unit: Hertz (Hz) = 1/s.
• Formula: $f=\frac{1}{T}$, where $T$ = time period (time for one complete wave).

3. Amplitude (A)

• Maximum displacement from the equilibrium position.
• Determines the energy of the wave (higher amplitude = more energy).

4. Wave Speed (v)

• Distance traveled by a wave per unit time.
• Formula: $v=f\cdot \lambda$ (speed = frequency × wavelength).
• Unit: m/s.

Visual:

Wave Speed in Different Media

The speed of a wave depends on the medium:

1. String Waves:
   • $v=\sqrt{\frac{T}{\mu }}$, where:
     • $T$ = tension in the string (N).
     • $\mu$ = linear mass density (kg/m).
2. Sound Waves in Air:
   • $v=331+0.6T$ m/s (where $T$ = temperature in °C).
   • At 20°C, $v\approx 343$ m/s.
3. Water Waves:
   • Depends on depth and gravity.

Example 1: Calculating Wave Speed on a String A string has a tension of 10 N and a linear mass density of 0.01 kg/m. Find the wave speed. Solution: $$v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{10}{0.01}}=\sqrt{1000}\approx 31.6\text{m/s}$$

Energy and Power of Waves

• Energy (E): Proportional to the square of the amplitude ($E\propto A^2$).
• Power (P): Energy transmitted per unit time.
  • For a wave on a string: $P=\frac{1}{2}\mu {\omega }^2{A}^{2}v$, where $\omega =2\pi f$.

Example 2: Comparing Energy If a wave’s amplitude doubles, how does its energy change? Solution: Since $E\propto A^2$, doubling $A$ increases $E$ by $2^2=4$ times.

Superposition and Interference

When two waves meet, they superpose (combine). The result depends on their phases:

1. Constructive Interference: Waves in phase → larger amplitude.
2. Destructive Interference: Waves out of phase → cancel out (if amplitudes are equal).

Visual:

![constructive and destructive interference diagram](/media/d380a4d12ae1969d946a.png "Shows two waves combining. (Image: CC BY-SA 3.0, via Wikimedia Commons)")

Standing Waves

Occur when two waves of the same frequency, amplitude, and wavelength travel in opposite directions and interfere.

• Nodes: Points with zero amplitude (no movement).
• Antinodes: Points with maximum amplitude.
• Fundamental Frequency (f₁): Lowest frequency for standing waves.
  • For a string fixed at both ends: $f_1=\frac{v}{2L}$, where $L$ = length of the string.

Example 3: Standing Waves on a String A string of length 1 m has a wave speed of 200 m/s. Find the fundamental frequency. Solution: $$f_1=\frac{v}{2L}=\frac{200}{2\times 1}=100\text{Hz}$$

Doppler Effect (Brief Introduction)

• Change in frequency (and wavelength) due to the relative motion of the source and observer.
• Formula:
  • If the observer moves: $f^{\prime }=f\left(\frac{v\pm v_o}{v}\right)$.
  • If the source moves: $f^{\prime }=f\left(\frac{v}{v\pm v_s}\right)$.
  • $v$ = wave speed, $v_o$ = observer speed, $v_s$ = source speed.
  • Use + when moving toward each other, – when moving apart.

Example 4: Doppler Effect A train blows a whistle of frequency 500 Hz while moving at 20 m/s toward a stationary observer. Speed of sound = 340 m/s. Find the observed frequency. Solution: $$f^{\prime }=f\left(\frac{v}{v-v_s}\right)=500\left(\frac{340}{340-20}\right)=500\times \frac{340}{320}=531.25\text{Hz}$$

Applications of Mechanical Waves

1. Music Instruments: Strings (guitar), air columns (flute), and membranes (drums) produce standing waves.
2. Medical Imaging: Ultrasound uses high-frequency sound waves to create images.
3. Seismology: Studying earthquake waves (P-waves = longitudinal, S-waves = transverse).
4. Communication: Radio waves (though electromagnetic, the concept applies to signal transmission).

Exam Tip

1. Memorize Key Formulas:

   • $v=f\lambda$
   • $v=\sqrt{\frac{T}{\mu }}$ (for strings)
   • $f_1=\frac{v}{2L}$ (fundamental frequency)
   • Doppler effect formulas.

2. Understand Concepts, Not Just Formulas:

   • Know the difference between transverse and longitudinal waves.
   • Visualize standing waves (nodes and antinodes).

3. Practice Problems:

   • NEB often asks for calculations involving wave speed, frequency, or Doppler effect.
   • Draw diagrams for standing waves and interference.

4. Common Mistakes to Avoid:

   • Confusing frequency and period ($f=\frac{1}{T}$).
   • Forgetting units (Hz for frequency, m/s for speed).
   • Misapplying Doppler effect signs (+ or –).

NEB-Style Questions

Short Answer (5 marks)

1. Define wavelength and frequency. A wave has a speed of 200 m/s and a frequency of 50 Hz. Calculate its wavelength. Answer:

   • Wavelength ($\lambda$) = distance between two consecutive crests.
   • Frequency ($f$) = number of waves per second.
   • $\lambda =\frac{v}{f}=\frac{200}{50}=4\text{m}$.

2. Differentiate between transverse and longitudinal waves with examples. Answer:

   Feature	Transverse Waves	Longitudinal Waves
   Particle Motion	Perpendicular	Parallel
   Example	Waves on a string	Sound waves

Long Answer (10 marks)

3. Explain the formation of standing waves in a string fixed at both ends. Derive the expression for the fundamental frequency. Answer:

   • Standing waves form when two identical waves travel in opposite directions and interfere.
   • Nodes (no displacement) and antinodes (maximum displacement) are created.
   • For a string of length $L$, the fundamental frequency is: $$f_1=\frac{v}{2L}$$ where $v$ is the wave speed.

4. A source emits sound waves of frequency 440 Hz. An observer moves toward the source at 10 m/s. If the speed of sound is 340 m/s, calculate the observed frequency. Answer: $$f^{\prime }=f\left(\frac{v+v_o}{v}\right)=440\left(\frac{340+10}{340}\right)=440\times 1.029\approx 453\text{Hz}$$

Final Note: Mechanical waves are everywhere—from music to earthquakes! Master the formulas, draw diagrams, and practice calculations to score well in NEB exams. 🚀