Mechanical Waves – Propagation, Energy & Superposition Explained | Chapter 15 of University Physics

Chapter 15 explores how mechanical waves transport energy through media without bulk matter motion. From transverse and longitudinal disturbances to standing wave patterns, this chapter provides the tools to model, analyze, and apply wave behavior across strings, air, and fluids.

Watch the full video summary here for detailed derivations and real-world examples.

Types of Mechanical Waves

• Transverse waves – Particle motion ⟂ wave direction (e.g., waves on strings).
• Longitudinal waves – Particle motion ∥ wave direction (e.g., sound in air).
• Surface waves – Combine transverse and longitudinal motion (e.g., water waves).

Wave Parameters & Periodic Waves

Key parameters describe any periodic wave:

• Amplitude (A): Maximum displacement from equilibrium.
• Wavelength (λ): Distance between repeating points.
• Frequency (f): Cycles per second, f = 1/T.
• Period (T): Time for one full cycle.
• Wave speed (v): v = λ f.

Wave Function & Wave Equation

A sinusoidal wave traveling in +x direction is modeled by:

y(x, t) = A cos(k x – ω t), where k = 2π/λ and ω = 2π f. These satisfy the one-dimensional wave equation: ∂²y/∂x² = (1/v²) ∂²y/∂t².

Wave Speed on a String

For a string under tension F and linear mass density μ:

v = √(F/μ). Tighter tension or lighter string increases propagation speed.

Energy Transport by Waves

Waves carry energy without net mass transport. The average power for a sinusoidal wave on a string is:

P_avg = ½ μ v ω² A². In three-dimensional waves, intensity obeys the inverse-square law: I ∝ 1/r².

Wave Superposition & Interference

The principle of superposition states overlapping waves add algebraically, producing:

• Constructive interference: amplitudes reinforce.
• Destructive interference: amplitudes cancel.

Reflections at boundaries invert or preserve phase depending on fixed or free ends.

Standing Waves & Normal Modes

Standing waves form when incident and reflected waves interfere in a fixed medium. Key features:

• Nodes: points of zero displacement.
• Antinodes: points of maximum displacement.
• For a string of length L fixed at both ends: λ_n = 2L/n, f_n = n (v/2L), where n is the mode number.

Musical instruments exploit these harmonics to produce distinct tones.

Conclusion

Mechanical waves underpin phenomena from musical acoustics to seismic events. By mastering wave parameters, equations, and interactions like superposition and standing modes, you gain powerful tools to predict and harness wave behavior in engineering and science.

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