2.6 Periodic Motion, Wave Fundamentals, Superposition & Standing Waves

2.6 Periodic Motion, Wave Fundamentals, Superposition & Standing Waves

Acoustics and mechanical wave mechanics describe how oscillatory energy propagates through compressible media. On the MCAT, wave phenomena bridge fundamental physics and clinical medicine: auditory transduction in the cochlea, acoustic resonance in the ear canal, tissue attenuation during sonography, and Doppler echocardiography of stenotic heart valves. Mastering wave fundamentals requires understanding wave speed medium-dependence, standing wave boundary conditions, decibel math, and the Doppler frequency equation.

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Wave Fundamentals & Periodic Motion

Simple Harmonic Motion (SHM)

Simple harmonic motion occurs when a displaced particle experiences a restoring force directly proportional to its displacement from equilibrium, directed opposite to that displacement:

$$F = -k x$$

• Mass-spring system: Period $T = 2\pi \sqrt{\frac{m}{k}}$, angular frequency $\omega = \sqrt{\frac{k}{m}}$
• Simple pendulum: Period $T = 2\pi \sqrt{\frac{L}{g}}$, angular frequency $\omega = \sqrt{\frac{g}{L}}$

Fundamental Wave Parameters

1. Wavelength ($\lambda$): Spatial distance between two consecutive in-phase points (crest to crest or compression to compression) (m).
2. Frequency ($f$): Number of complete oscillations or cycles passing a fixed point per unit time (Hertz, $1\text{ Hz} = 1\text{ s}^{-1}$).
3. Period ($T$): Time required to complete one full cycle ($T = 1/f$) (s).
4. Amplitude ($A$): Maximum displacement from equilibrium. Wave energy and intensity scale with the square of amplitude ($I \propto A^2$).
5. Wave Speed ($v$): The propagation speed of the wave disturbance through space: $$v = f \lambda = \frac{\lambda}{T}$$

Medium-Dependence of Wave Speed

The Cardinal Rule of Wave Mechanics: Wave speed ($v$) is determined strictly by the physical properties of the transmitting medium, not by the wave's frequency or amplitude. For longitudinal sound waves propagating through a bulk compressible fluid or tissue:

$$v = \sqrt{\frac{B}{\rho}}$$

where $B$ is the bulk modulus (incompressible stiffness/elasticity) and $\rho$ is the density of the medium.

Although solids and liquids are denser than gases, their bulk moduli are orders of magnitude greater ($B_{\text{solid}} \gg B_{\text{liquid}} \gg B_{\text{gas}}$). Consequently, sound travels fastest in stiff solids, intermediate in liquids, and slowest in compressible gases:

• Air ($20^\circ\text{C}$): $v \approx 343\text{ m/s}$
• Pure Water ($20^\circ\text{C}$): $v \approx 1480\text{ m/s}$
• Human Soft Tissue: $v \approx 1540\text{ m/s}$ (universal calibration standard for diagnostic ultrasound)
• Cortical Bone: $v \approx 3500 - 4000\text{ m/s}$

Boundary Transitions

When a wave crosses an interface between two different media:

1. Frequency ($f$) remains constant because oscillations are driven by the source.
2. Wave speed ($v$) changes according to the mechanical properties of the new medium.
3. Wavelength ($\lambda$) changes in direct proportion to speed: $\lambda = v / f$.

Transverse vs. Longitudinal Waves

• Transverse Waves: Particle oscillations are perpendicular to the direction of wave propagation (e.g., electromagnetic radiation, surface water waves, plucking a guitar string). Transverse waves can be polarized.
• Longitudinal Waves: Particle oscillations are parallel to the direction of wave propagation, creating alternating regions of compression (high density/pressure) and rarefaction (low density/pressure) (e.g., sound waves, diagnostic ultrasound pulses). Longitudinal waves cannot be polarized.

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Superposition, Interference & Standing Waves

Principle of Superposition

When two or more waves travel through the same medium simultaneously, the net resultant displacement at any coordinate is the algebraic vector sum of individual displacements:

• Constructive Interference: Waves arrive in phase (phase difference $\Delta\phi = 0, 2\pi, 4\pi$; path length difference $\Delta d = m \lambda$, where $m$ is an integer). Amplitudes add constructively ($A_{\text{net}} = A_1 + A_2$).
• Destructive Interference: Waves arrive out of phase (phase difference $\Delta\phi = \pi, 3\pi, 5\pi$; path length difference $\Delta d = [m + \frac{1}{2}]\lambda$). Amplitudes subtract ($A_{\text{net}} = |A_1 - A_2|$).

Standing Waves

A standing wave forms when two identical harmonic waves traveling in opposite directions interfere within a bounded medium. Standing waves exhibit fixed spatial points of zero oscillation (nodes) and points of maximal oscillatory excursion (antinodes). The distance between any two adjacent nodes (or two adjacent antinodes) is exactly half a wavelength ($\frac{1}{2} \lambda$).

1. Open Pipes & Fixed Strings (Symmetric Boundaries)

An air pipe open at both ends supports displacement antinodes at both ends. A string clamped at both ends supports displacement nodes at both ends. Both systems share identical harmonic equations:

$$\lambda_n = \frac{2L}{n}, \quad f_n = \frac{n v}{2L} = n f_1 \quad (n = 1, 2, 3, 4, \dots)$$

where $n$ is the harmonic number, $L$ is conduit length, and $f_1 = \frac{v}{2L}$ is the fundamental frequency. All integer harmonics exist.

2. Closed-Open Pipes (Asymmetric Boundaries)

An air column closed at one end and open at the other (such as a vocal tract or ear canal) requires a node at the closed boundary (particles cannot oscillate against a rigid wall) and an antinode at the open boundary:

$$\lambda_n = \frac{4L}{n}, \quad f_n = \frac{n v}{4L} = n f_1 \quad (n = 1, 3, 5, 7, \dots)$$

MCAT High-Yield Rule: A closed-open pipe supports only odd harmonics ($n = 1, 3, 5, \dots$). Even harmonics ($n = 2, 4, 6, \dots$) cannot physically exist because they cannot satisfy the simultaneous node/antinode boundary conditions.

Biological Correlate: External Auditory Canal Resonance

The human external auditory canal is an air-filled tube bounded externally by the pinna (open) and internally by the tympanic membrane (closed). Average canal length is $L \approx 2.5\text{ cm} = 0.025\text{ m}$. Assuming speed of sound in air $v = 343\text{ m/s}$:

$$f_1 = \frac{v}{4L} = \frac{343\text{ m/s}}{4(0.025\text{ m})} = \frac{343}{0.10} = 3430\text{ Hz} \approx 3.4\text{ kHz}$$

This acoustic resonance amplifies sound waves in the $2 - 4\text{ kHz}$ range by up to $10 - 15\text{ dB}$, precisely corresponding to human speech frequencies and explaining peak auditory sensitivity.

Summary of Standing Wave Resonators and Harmonics

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