Module 6: Waves and Sound

Introduction to Waves and Sound

Welcome to Module 6: Waves and Sound, a vital chapter in MCAT physics that links foundational wave mechanics with hearing, communication, and diagnostic technologies. This lesson explores how energy travels through space and matter via oscillations—whether in the form of ripples on water, vibrations in a string, or pressure fluctuations in air that we perceive as sound.

On the MCAT, wave principles are central to understanding sound transmission in the ear, ultrasound imaging, Doppler effects in blood flow measurement, and even the behavior of light and electromagnetic radiation in later topics. Mastery of wave characteristics like frequency, wavelength, speed, and amplitude will allow you to analyze how physical systems carry energy, how interference shapes patterns, and how sound behaves in biological and clinical contexts.

Learning Objectives:

By the end of this section, you should be able to:

Define what a wave is and explain how it transfers energy without transferring matter.
Differentiate between transverse and longitudinal waves with examples.
Use and manipulate fundamental wave equations.
Identify the roles of amplitude, wavelength, frequency, period, and speed in wave behavior.
Understand how wave speed depends on medium properties (e.g., tension, mass, density, bulk modulus).
Explain the role of surface tension in the formation of capillary waves and how it relates to wave propagation in fluids.

Why Waves and Sound Matters on the MCAT

Waves and sound aren’t just abstract physics concepts — they show up directly in MCAT-relevant biology and clinical applications. From how we hear and speak to how physicians interpret heartbeats, breathing patterns, and imaging data, wave behavior underlies core human functions and modern diagnostic tools.

You are expected to:

Analyze how the ear converts sound waves into neural signals.
Interpret graphs of standing waves in strings, pipes, and biological tubes.
Understand how the Doppler effect explains changes in pitch and moving blood flow.
Apply wave equations and harmonics to model auditory systems and clinical instrumentation (e.g., ultrasound).

While waves and sound account for about 5–10% of MCAT physics questions, their applications cut across disciplines — linking physical science to sensory biology, speech, and noninvasive imaging — making them high-yield beyond their raw content weight.

Module Overview:

Wave Properties and Types
Standing Waves and Harmonics
Sound Properties: Pitch, Loudness, Intensity, and the Decibel Scale
The Doppler Effect and Clinical Applications
Ultrasound Imaging: Frequency, Resolution, and Time-of-Flight
Shock Waves and Supersonic Motion

Wave Properties and Types

What Is a Wave?

A wave is a dynamic physical phenomenon in which disturbances or oscillations propagate through a medium or space, transporting energy from one location to another without permanent displacement of the medium itself. In wave motion, particles of the medium experience oscillations, but the overall medium remains in place.

This distinction, energy transfer without mass transfer, is central to understanding wave behavior.

There are two broad categories:

Mechanical waves: Require a material medium (e.g., sound waves, water waves).
Electromagnetic waves: Can propagate through vacuum (e.g., light, radio waves).

Fundamental Quantities That Describe a Wave

Let’s begin with the five fundamental quantities used to describe any wave, whether mechanical or electromagnetic.

Quantity	Symbol	Definition	Units (SI)
Wavelength	λ	Distance between identical points on consecutive cycles (e.g., crest-to-crest)	meters (m)
Frequency	f	Number of complete cycles per second	Hertz (Hz = s^-1)
Period	T	Time taken for one complete cycle	seconds (s)
Amplitude	A	Maximum displacement from equilibrium	meters (m)
Wave speed	v	Rate at which the disturbance propagates	m/s
Node	None	Point along a standing wave where there is no displacement — the medium remains stationary at all times.	None
Antinode	None	Point of maximum displacement on a standing wave — where the medium oscillates with greatest amplitude.	None

Relationships Between Variables

These quantities are related by two foundational equations:

Frequency and period are reciprocals:

$$
T = \frac{1}{f} \quad \text{and} \quad f = \frac{1}{T}
$$

Wave speed is the product of frequency and wavelength:

$$
v = f \lambda
$$

This equation applies to all wave types. But the speed v itself depends not on frequency, but on the properties of the medium (discussed below).

Wave Classification by Motion and Medium

By Direction of Particle Motion:

Transverse Waves

Particles move perpendicular to the wave’s direction of travel.
Ex: Waves on a string, ripples on water, electromagnetic waves.

Visualize a rope flicked vertically – wave travels horizontally, but rope oscillates up and down.

Longitudinal Waves

Particles move parallel to the wave’s direction of travel.
Ex: Sound waves, compression waves in a slinky, seismic P-waves.

Visualize pushing and pulling a slinky – regions of compression and rarefaction move along the same axis as oscillation.

By Requirement of Medium:

Mechanical Waves

Require a medium to propagate.
Cannot travel through vacuum.
Examples: Sound, water, seismic waves.

Electromagnetic Waves

Do not require a medium.
Propagate through space at the speed of light.
Examples: Radio, infrared, visible light, x-rays, gamma rays.

Wave Speed and Medium Properties

Wave speed depends on the medium’s elastic and inertial properties:

For strings and solids, wave speed is governed by:

$$
v = \sqrt{\frac{T}{\mu}}
$$

Where:

$$
v = \text{Wave speed (m/s)}
$$

$$
T = \text{Tension in the string (Newtons, N)}
$$

$$
\mu = \text{Linear mass density (kg/m)}
$$

For fluids, wave speed is:

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

Where:

$$
v = \text{Wave speed in the fluid (m/s)}
$$

$$
B = \text{Bulk modulus of the fluid (Pa), a measure of compressibility}
$$

$$
\rho = \text{Density of the fluid (kg/m}^3\text{)}
$$

Physical Insight: Waves move faster in materials that are more rigid (higher B) and less massive (lower ρ). That’s why sound travels faster in steel than in air.

Real-Life Illustrations

Scenario	Type of Wave	Description
String instrument (e.g., violin)	Transverse	Vibrating string generates transverse standing wave, transmitted to air as sound.
Human speech	Longitudinal	Vocal cords compress air, creating sound waves propagating to listener
Light through vacuum	Electromagnetic	Transverse wave of electric and magnetic fields; no medium needed
Ultrasound in tissue	Longitudinal	High-frequency sound waves reflecting from tissue boundaries

Key Takeaways and Concept Mastery

Wave speed depends on the medium, not on wave properties like amplitude or frequency. Frequency can change if the wave enters a new medium, but speed is dictated by the medium.
Transverse vs. longitudinal distinction is fundamental. Most confusion stems from forgetting which direction the medium vibrates in relation to wave motion.
Waves carry energy, not mass. Even tsunami waves don’t carry the ocean with them – they transfer energy across massive distances.
Energy and amplitude are related quadratically: Energy ∝ A². This means that doubling amplitude increases energy transmission fourfold.

Standing Waves and Harmonics

What Are Standing Waves?

A standing wave arises when two waves of the same frequency, amplitude, and speed travel in opposite directions and interfere with one another.

Unlike traveling waves, standing waves do not propagate energy across the medium. Instead, they exhibit fixed points of no displacement (nodes) and maximum oscillation (antinodes).
These waveforms oscillate in place, making them crucial in physical systems like musical instruments, vocal cords, and biomedical waveguides.

The Physics of Superposition

Superposition principle: When two or more waves overlap in space, the resultant displacement is the algebraic sum of the individual displacements.

Constructive interference: Peaks align, resulting in higher amplitude
Destructive interference: Peaks and troughs cancel, resulting in node formation

In a standing wave:

At nodes, destructive interference is perfect, resulting in no motion
At antinodes, constructive interference is perfect, resulting in maximal motion.

Standing Waves on a String (Both Ends Fixed)

Physical Setup:

A string fixed at both ends can only support waveforms that “fit” the string length exactly. The boundaries must be nodes (no motion).

This constraint allows only specific wavelengths.

Allowed Wavelengths:

$$
\lambda_n = \frac{2L}{n}
$$

$$
f_n = \frac{n v}{2L}
$$

Where:

$$
\lambda_n = \text{Wavelength of the } n^\text{th} \text{ harmonic (m)}
$$

$$
n = \text{Harmonic number (1, 2, 3, \ldots)}
$$

$$
L = \text{Length of the string (m)}
$$

Each n represents a harmonic:

1st harmonic (fundamental): one anti-node in center
2nd harmonic: two half-wavelengths
3rd harmonic: three half-wavelengths, etc.

Allowed Frequencies:

$$
f_n = \frac{v}{\lambda_n} = \frac{n v}{2L}
$$

Where v is the speed of the wave on the string (depends on tension and mass per unit length):

$$
v = \sqrt{\frac{T}{\mu}}
$$

Where:

$$
v = \text{Wave speed on the string (m/s)}
$$

$$
T = \text{Tension in the string (Newtons, N)}
$$

$$
\mu = \text{Linear mass density (kg/m)}
$$

This equation is crucial for understanding how changing string tension (like tightening a guitar string) affects wave speed and thus pitch.

Standing Waves in Open Pipes (Both Ends Open)

Physical Setup:

Air columns with open ends (like flutes) must have antinodes at both ends, as air is free to oscillate.

Allowed Wavelengths:

$$
\lambda_n = \frac{2L}{n}
$$

Where:

$$
\lambda_n = \text{Wavelength of the } n^\text{th} \text{ harmonic (m)}
$$

$$
L = \text{Length of the pipe (m)}
$$

$$
n = \text{Harmonic number (1, 2, 3, \ldots)}
$$

It’s the same pattern as for strings – open-open boundary conditions are mathematically identical.

Allowed Frequencies:

$$
f_n = \frac{n v}{2L}
$$

Where:

$$
f_n = \text{Frequency of the } n^\text{th} \text{ harmonic (Hz)}
$$

$$
n = \text{Harmonic number (1, 2, 3, \ldots)}
$$

$$
v = \text{Speed of sound in the pipe (m/s)}
$$

$$
L = \text{Length of the pipe (m)}
$$

Standing Waves in Closed Pipes (One End Closed)

Physical Setup:

In closed pipes (e.g., clarinets, human vocal tract), the closed end must be a node (no air movement), and the open end must be an antinode.

Key Difference:

The pipe can only accomodate odd harmonics, because a half-integer number of wavelengths doesn’t fit with node-antinode boundaries.

Allowed Wavelengths:

$$
\lambda_n = \frac{4L}{n}, \quad n = 1, 3, 5, \ldots
$$

Where:

$$
\lambda_n = \text{Wavelength of the } n^\text{th} \text{ harmonic (m)}
$$

$$
L = \text{Length of the pipe (m)}
$$

$$
n = \text{Harmonic number — only odd integers (1, 3, 5, \ldots)}
$$

Allowed Frequencies:

$$
f_n = \frac{n v}{4L}, \quad n = 1, 3, 5, \ldots
$$

Where:

$$
f_n = \text{Frequency of the } n^\text{th} \text{ harmonic (Hz)}
$$

$$
n = \text{Harmonic number — only odd integers (1, 3, 5, \ldots)}
$$

$$
v = \text{Speed of sound in the pipe (m/s)}
$$

$$
L = \text{Length of the pipe (m)}
$$

This pattern arises because the closed end is always a node, and the open end is always an antinode, so only odd harmonics can form standing waves in this system.

Common Pitfall:

Closed pipes cannot support even-numbered harmonics.
1st harmonic = one-quarter of a full wavelength.
3rd harmonic = three-quarters of a full wavelength.

System	End Conditions	Harmonics Allowed
String	Node – Node	All harmonics (n = 1, 2, 3, …)
Open pipe	Antinode – Antinode	All harmonics (n = 1, 2, 3, …)
Closed pipe	Node – Antinode	Odd harmonics only (n = 1, 3, 5 …)

Physiological and Instrumental Applications

Human Vocal Tract

Acts like a closed pipe: vocal folds at base (node), mouth at top (antinode).
Explains resonant frequencies of speech (formants) and why vowels have different acoustic patterns.

String Instruments

Guitar strings exhibit standing wave harmonics based on string length, tension, and boundary conditions.
Musicians tune pitch by adjusting tension (changes v) or finger placement (changes L).

Key Mastery Concepts

Standing waves require boundary conditions that enforce node/antinode patterns.
Only certain wavelengths “fit” within a given system – these are the allowed harmonics.
Frequency is quantized in these systems, only specific values are possible.
Closed-end systems (e.g., vocal tract, clarinet) inherently limit harmonic content.
Wave speed is constant in a given medium; changes in harmonic number only change frequency and wavelength, not speed.

Sound Properties (Pitch, Loudness, Doppler Effect)

What is Sound?

Sound is a mechanical, longitudinal wave consisting of compressions and rarefactions in a medium such as air, water, or solid tissue.

Unlike transverse waves, the oscillations of particles in sound waves are parallel to the wave’s direction of travel.
Sound cannot propagate in vacuum – it requires a medium to transmit pressure disturbances.

Wave Equation for Sound:

v = f \lambda

Where:

v = \text{Speed of sound in the medium (m/s)}

f = \text{Frequency of the sound wave (Hz)}

\lambda = \text{Wavelength of the sound wave (m)}

In air at room temperature, the speed of sound is approximately 343 m/s

Pitch vs. Frequency

Frequency (f):

Number of oscillation cycles per second.
Measured in Hertz (Hz).
Physical property of the wave.

Pitch:

Subjective perception of frequency.
Higher frequency results in higher pitch (e.g., piccolo).
Lower frequency results in lower pitch (e.g., tuba).

Tip: Frequency is objective. Pitch is perceptual. The two correlate but are not synonymous.

Loudness vs. Intensity

Amplitude:

The maximum displacement of the medium particles from equilbrium.
Proportional to wave energy: E ∝ A²

Intensity (I):

Power per unit area (W/m²).

I = \frac{P}{A}

Where:

I = \text{Intensity of the wave (Watts/m}^2\text{)}

P = \text{Power delivered by the wave (Watts, W)}

A = \text{Area over which the power is spread (m}^2\text{)}

Describes the rate at which energy is delivered over a surface
Inversely proportional to distance squared (spherical spreading):

I \propto \frac{1}{r^2}

Where:

I = \text{Intensity of the sound wave (W/m}^2\text{)}

r = \text{Distance from the source (m)}

MCAT Analogy: Think of sound as a flashlight beam – closer is brighter (louder), farther is dimmer (quieter).

Loudness:

Subjective perception of sound intensity.
Influenced by both amplitude and frequency sensitivity of the human ear.

The Decibel (dB) Scale

The decibel scale is logarithmic and reflects how humans perceive loudness.

Sound Level (β):

\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)

Where:

\beta = \text{Sound level (decibels, dB)}

I = \text{Intensity of the sound (W/m}^2\text{)}

I_0 = \text{Reference intensity } (10^{-12} \text{ W/m}^2)

Important Logarithmic Effects:

A +10 dB increase = 10 x intensity
A +20 dB increase = 100x intensity
A -10 dB decrease = 1/10^th the intensity

Watch Out: Many premeds incorrectly assume decibels are linear. They are non – 10 dB = 10x, not 2x.

Doppler Effect

The Doppler effect is the perceieved change in sound frequency when either the source or observer (or both) are moving relative to one another.

Doppler Equation

f’ = f \cdot \frac{v \pm v_o}{v \mp v_s}

Where:

f’ = \text{Observed frequency (Hz)}

f = \text{Actual emitted frequency (Hz)}

v = \text{Speed of sound in the medium (m/s)}

v_o = \text{Speed of the observer (m/s)}

v_s = \text{Speed of the source (m/s)}

Sign Conventions:

Use top signs when the observer and source are moving toward each other.
Use bottom signs when moving away.

Conceptual Summary:

Relative Motion	Effect on Observed Frequency f’
Source and observer move toward each other	f’ > f (higher pitch)
Source and observer move apart	f’ < f (lower pitch)

Key Takeaways and Mastery Points

Sound is a longitudinal mechanical wave, dependent on the medium.
Pitch is the perceptual correlate of frequency; loudness is the perceptual correlate of intensity.
Sound intensity falls off with the square of distance: I ∝ 1/r².
Decibel scale is logarithmic: every +10 dB = x10 intensity.
Doppler effect shifts observed frequency depending on motion of observer/source.
Clinical imaging techniques like Doppler Ultrasound rely on sound wave physics for real-time measurement.

Ultrasound and Shock Waves

What is Ultrasound?

Ultrasound refers to sound waves with frequencies > 20,000 Hz – well above the range of human hearing (20 – 20,000 Hz).

In medical imaging, typical ultrasound frequencies range from 2MHz to 15 MHz.
It is a non-ionizing, real-time, and safe imaging modality commonly used in obstetrics, cardiology, and vascular medicine.

Wave Physics of Ultrasound

Ultrasound obeys the same wave principles you’ve already learned:

It is a longitudinal mechanical wave.
It requires a medium and propagates at a speed determine by:

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

Where:

$$
v = \text{Speed of ultrasound in the medium (m/s)}
$$

$$
B = \text{Bulk modulus of the medium (Pa)}
$$

$$
\rho = \text{Density of the medium (kg/m}^3\text{)}
$$

Trade-Off Between Resolution and Penetration

High Frequency (10 – 15 MHz)	Low Frequency (2 – 5 MHz)
Shorter wavelength	Longer wavelength
Better resolution	Better penetration
Used for superficial imaging (e.g., thyroid, breast)	Used for deeper structures (e.g., liver, fetus)

MCAT Insight: The higher the frequency, the better the detail – but shorter wavelengths are absorbed faster, reducing depth penetration.

Reflection and Acoustic Impedance

Ultrasound waves reflect when the hit boundaries between tissues of different acoustic impedance. It is not necessary to know the equation for this concept for the MCAT, but having a conceptual understanding may be helpful on the MCAT.

The greater the difference in impedance, the greater the reflection.
Reflected waves return to the probe and are converted into an image.
Interfaces like soft tissue vs. bone or soft tissue vs. air reflect strongly, resulting in poor transmission beyond.

Clinical relevance: Air and bone interfere with ultrasound. That’s why gel is used – to eliminate air gaps between probe and skin.

Key Mastery Concepts

Ultrasound uses high-frequency sound waves to image soft tissues based on reflected echoes.
Resolution increases with frequency, but depth penetration decreases.
Acoustic impedance mismatch determines reflection intensity.
Doppler ultrasound measures blood velocity based on frequency shift.