Mechanical Waves

A mechanical wave is a disturbance that propagates through an elastic medium carrying energy without net transport of matter. Dropping a stone into a pond creates ripples that travel outward: each water molecule oscillates vertically around its equilibrium position but does not travel horizontally. The stone's energy propagates; the matter does not.

Why does a mechanical wave need a medium? Because the disturbance is transmitted through interactions between adjacent particles. In a vacuum, mechanical waves cannot exist. Light (electromagnetic waves) is different: it can travel through vacuum because it is a self-sustaining oscillation of electric and magnetic fields.

Wave types by oscillation direction:

• Transverse waves: particles oscillate perpendicular to the direction of propagation. Classic example: a taut string shaken vertically produces a wave that travels horizontally. Electromagnetic waves, earthquake S-waves, and waves on membranes are transverse.
• Longitudinal waves: particles oscillate parallel to propagation. Sound is the classical example: a loudspeaker membrane alternately compresses and rarefies the air, and these high/low pressure regions travel toward the listener. Earthquake P-waves and slinky compression waves are longitudinal.

Real-world example — earthquakes: Earthquakes generate both types. P-waves (primary, longitudinal) travel faster (~6 km/s in the crust) and arrive first. S-waves (secondary, transverse) are slower (~3.5 km/s) but carry more energy and cause most of the damage. By measuring the P-S delay at different seismic stations, geologists locate the epicenter.

Key quantities:

• Amplitude $A$ (m): maximum displacement from equilibrium. Energy transported is proportional to $A^2$: waves with double amplitude carry 4 times the energy.
• Period $T$ (s): time for one complete oscillation.
• Frequency $f = 1/T$ (Hz): number of oscillations per second.
• Wavelength $\lambda$ (m): distance between two consecutive crests (or troughs, or any two in-phase points).
• Angular frequency $\omega = 2\pi f = 2\pi/T$ (rad/s): angular speed of the associated circular motion.
• Wave number $k = 2\pi/\lambda$ (rad/m): how many spatial oscillations in $2\pi$ meters.

Fundamental relation $v = \lambda f$: wave speed is determined by medium properties (tension, density, elasticity). If speed is fixed, frequency and wavelength are inversely proportional: higher frequency means shorter wavelength, and vice versa.

We start from simple harmonic motion: a point mass oscillates in time as $y(t) = A\cos(\omega t + \phi_0)$. To describe a wave we must add spatial dependence: two different points along the propagation direction are at different phases of the same periodic motion.

Constructing the wave function: Imagine taking a photograph of the wave at a fixed time $t$. We see a sinusoid in space: $y(x) = A\cos(kx + \text{constant})$. The wave number $k = 2\pi/\lambda$ tells us how many spatial oscillations fit in $2\pi$ meters. Now, if the wave travels to the right, the crest that was at $x$ at $t=0$ will have moved to $x + vt$ at time $t$. Mathematically: what we "saw" at $(x,0)$ we now see at $(x+vt, t)$. Rewriting: $y(x,t) = A\cos(k(x - vt) + \phi_0) = A\cos(kx - kvt + \phi_0)$. Since $kv = (2\pi/\lambda)(\lambda f) = 2\pi f = \omega$, we obtain:

$y(x,t) = A\cos(kx - \omega t + \phi_0)$

This is the progressive harmonic wave equation (rightward). For a leftward wave: $y = A\cos(kx + \omega t + \phi_0)$.

Parameter interpretation: $kx - \omega t$ is the phase. Fix a crest: $kx - \omega t = \text{constant} \Rightarrow x = (\text{constant} + \omega t)/k$. The crest moves with speed $dx/dt = \omega/k = v$, called the phase velocity.

Wave differential equation: From $y(x,t) = f(x-vt)$ (where $f$ is any function, not just sinusoidal) we can derive the partial differential equation that all waves satisfy:

$\dfrac{\partial^2 y}{\partial t^2} = v^2 \dfrac{\partial^2 y}{\partial x^2}$

Verification for $y = A\cos(kx - \omega t)$:

• $\partial y/\partial t = \omega A \sin(kx - \omega t) \Rightarrow \partial^2 y/\partial t^2 = -\omega^2 A \cos(kx - \omega t) = -\omega^2 y$
• $\partial y/\partial x = -k A \sin(kx - \omega t) \Rightarrow \partial^2 y/\partial x^2 = -k^2 A \cos(kx - \omega t) = -k^2 y$
• Therefore $\partial^2 y/\partial t^2 = (\omega^2/k^2) \partial^2 y/\partial x^2 = v^2 \partial^2 y/\partial x^2$ ✓

Physical meaning: the acceleration of a point ($\partial^2 y/\partial t^2$) is proportional to the curvature of the string ($\partial^2 y/\partial x^2$). Where the string is more curved, the net restoring force is greater and the acceleration is stronger — exactly as expected from an elastic medium.

The speed of a mechanical wave does NOT depend on frequency or amplitude, but only on the physical properties of the medium: elasticity (ability to transmit the restoring force) and inertia (resistance to motion).

Wave on a taut string:
$v = \sqrt{\dfrac{F_T}{\mu}}$
where $F_T$ is the string tension (N) and $\mu = m/L$ is the linear mass density (kg/m).

Physical intuition — the derivation: Consider a small string segment of length $\Delta x$. Tension acts tangentially at both ends. If the string is curved, the two tensions have opposite vertical components, generating a net restoring force. Applying Newton's second law ($F = ma$) and taking the limit $\Delta x \to 0$, we obtain the wave equation $\partial^2 y/\partial t^2 = (F_T/\mu) \cdot \partial^2 y/\partial x^2$, hence $v^2 = F_T/\mu$.

More tension → greater restoring force → the wave travels faster. Higher density → greater inertia → slower wave. A thicker guitar string (higher $\mu$) produces lower notes at the same tension and length.

Wave in a solid (thin bar):
$v = \sqrt{\dfrac{E}{\rho}}$
where $E$ is Young's modulus (material stiffness, in Pa) and $\rho$ is the mass density (kg/m³). Steel ($E \approx 200$ GPa, $\rho \approx 7800$ kg/m³) has $v \approx 5060$ m/s. Rubber has much lower speed because it is less stiff.

Sound in a gas:
$v = \sqrt{\dfrac{\gamma RT}{M}}$
where $\gamma = c_p/c_v$ is the heat capacity ratio (1.4 for air), $R = 8.314$ J/(mol·K) is the gas constant, $T$ is the temperature in kelvin, $M$ is the molar mass (kg/mol). In air at 20°C: $v \approx 343$ m/s.

Temperature dependence: $v \approx 331\sqrt{T/273} \;\mathrm{m/s}$. For each degree Celsius increase, speed rises by about 0.6 m/s. This is why sound travels faster on warm days.

Comparison across media:

• Air (20°C): 343 m/s
• Water (20°C): 1482 m/s
• Steel: ~5960 m/s
• Glass: ~5640 m/s

Sound travels much faster in solids than liquids, and faster in liquids than gases, because molecules are closer and interact more strongly.

The wave equation $\partial^2 y/\partial t^2 = v^2 \partial^2 y/\partial x^2$ is a linear equation: if $y_1$ and $y_2$ are solutions, so is $y_1 + y_2$. This is the superposition principle — waves add linearly without disturbing each other.

Interference of two waves with the same frequency:
Consider two waves traveling in the same direction with the same amplitude $A$, same angular frequency $\omega$, same wave number $k$, but different phase:
$y_1 = A\cos(kx - \omega t)$, $y_2 = A\cos(kx - \omega t + \Delta\phi)$

Using the sum-to-product identity $\cos\alpha + \cos\beta = 2\cos\frac{\alpha-\beta}{2}\cos\frac{\alpha+\beta}{2}$:

$y_{tot} = y_1 + y_2 = 2A \cos\frac{\Delta\phi}{2} \; \cos\left(kx - \omega t + \frac{\Delta\phi}{2}\right)$

The result is a wave at the same frequency with amplitude $A_{tot} = 2A|\cos(\Delta\phi/2)|$.

Limit cases:

• Constructive interference ($\Delta\phi = 0, 2\pi, 4\pi, \ldots$): $\cos(\Delta\phi/2) = \pm 1$, $A_{tot} = 2A$. The resulting wave has double amplitude and quadruple intensity ($I \propto A^2$).
• Destructive interference ($\Delta\phi = \pi, 3\pi, 5\pi, \ldots$): $\cos(\Delta\phi/2) = 0$, $A_{tot} = 0$. The two waves cancel completely.

Path difference and phase difference: If the phase difference arises from different path lengths $\Delta r$, then $\Delta\phi = (2\pi/\lambda)\Delta r$. Therefore:

• Constructive: $\Delta r = m\lambda$ (path difference is integer multiple of $\lambda$)
• Destructive: $\Delta r = (m+\tfrac{1}{2})\lambda$ (half-integer multiple)

Real-world example — noise-canceling headphones: Microphones capture ambient noise, electronics generate a wave with opposite phase ($\Delta\phi = \pi$), and the two waves interfere destructively at the ear, canceling the noise.

Beats — different frequencies: If two waves have slightly different frequencies ($\omega_1 \approx \omega_2$):
$y_{tot} = \left[2A\cos\left(\frac{\Delta\omega}{2}t\right)\right] \cos\left(\frac{\omega_1+\omega_2}{2}t\right)$
The amplitude oscillates slowly with frequency $f_{beat} = |f_2 - f_1|$, producing a "pulsing" sound. Musicians use beats to tune instruments: when the beat frequency drops to zero, the instrument is in tune.

A standing wave is created by the superposition of two identical waves traveling in opposite directions. Consider a wave on a string reflected at a fixed end: the incident and reflected waves interfere to create a profile that does not propagate but oscillates in place.

Mathematical derivation:
$y_1 = A\cos(kx - \omega t)$ (rightward), $y_2 = A\cos(kx + \omega t)$ (leftward, reflected)

Summing: $y = y_1 + y_2 = A[\cos(kx-\omega t) + \cos(kx+\omega t)] = 2A\cos(kx)\cos(\omega t)$

This is the standing wave form: the spatial dependence ($\cos(kx)$) and temporal dependence ($\cos(\omega t)$) are separated. Every point on the string oscillates at the same frequency $\omega$, but with amplitude $2A|\cos(kx)|$ that varies with position.

Nodes and antinodes:

• Nodes: points where $\cos(kx) = 0 \Rightarrow kx = (2n+1)\pi/2 \Rightarrow x = (2n+1)\lambda/4$. They never move.
• Antinodes: points where $|\cos(kx)| = 1 \Rightarrow kx = n\pi \Rightarrow x = n\lambda/2$. They oscillate with maximum amplitude $2A$.

Boundary conditions — string fixed at both ends: Both ends must be nodes (they cannot move). Therefore $x=0$ and $x=L$ must be nodes. This forces $\lambda_n = 2L/n$, with $n = 1,2,3,\ldots$. The resonance frequencies are:

$f_n = \dfrac{v}{\lambda_n} = \dfrac{n\,v}{2L} \quad (n=1,2,3,\ldots)$

$n=1$ is the fundamental frequency (first harmonic); $n=2$ is the second harmonic, and so on. Each harmonic has $n$ antinodes.

Open-closed pipe: The open end is an antinode (air can move freely), the closed end is a node. This only allows $\lambda_n = 4L/(2n-1)$, i.e. only odd harmonics:

$f_n = \dfrac{(2n-1)\,v}{4L} \quad (n=1,2,3,\ldots)$

Open-open pipe: Both ends are antinodes → same frequencies as the string: $f_n = nv/(2L)$.

Real-world examples:

• A guitar produces the fundamental note and its harmonics. The timbre ("warm" or "bright" sound) depends on the relative strength of the harmonics.
• A clarinet behaves as an open-closed pipe: it produces only odd harmonics, giving it a characteristic sound.
• The Tacoma Narrows Bridge (1940) entered resonance with the wind, accumulating energy until collapse — a dramatic example of mechanical resonance.

Sound is a mechanical longitudinal wave that propagates through a medium (solid, liquid, or gas) as a pressure oscillation. A sound source (vocal cord, loudspeaker, tuning fork) sets neighboring molecules into vibration, creating alternating regions of compression (high pressure) and rarefaction (low pressure) that travel through the medium.

Human hearing range: The human ear perceives frequencies between ~20 Hz and ~20,000 Hz (20 kHz). Infrasound (<20 Hz) and ultrasound (>20 kHz) exist but we cannot hear them. Dogs hear up to ~45 kHz, dolphins up to ~150 kHz.

Sound intensity $I$: power transported per unit area (W/m²). For a point source emitting isotropically (equally in all directions):

$I = \dfrac{P}{4\pi r^2}$

Intensity decreases with the square of distance: doubling the distance reduces intensity to one-fourth ($I \propto 1/r^2$). This is the inverse square law.

Decibel scale (dB): The human ear responds logarithmically to intensity. Instead of using $I$ directly, we use the sound level:

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

where $I_0 = 10^{-12}$ W/m² is the threshold of hearing (the faintest sound the human ear can detect at 1 kHz).

Examples of sound levels:

• 0 dB: Threshold of hearing ($I_0 = 10^{-12}$ W/m²)
• 30 dB: Whisper at 1 m
• 60 dB: Normal conversation
• 90 dB: Heavy traffic / vacuum cleaner
• 120 dB: Rock concert / jackhammer — threshold of pain
• 140 dB: Jet engine at 30 m — immediate hearing damage

Important properties of the dB scale:

• Doubling $I$ → $\beta$ increases by $10\log_{10}(2) \approx +3$ dB. Two identical speakers produce 3 dB more than one.
• Tenfold $I$ → $\beta$ increases by $+10$ dB. A sound 10 times more intense is perceived as roughly "twice as loud".
• Damage threshold: prolonged exposure above 85 dB can cause permanent hearing damage. This is why rock concerts are potentially dangerous.

Practical calculation: A 100 W source at 50 m produces $I = 100/(4\pi\cdot 50^2) = 3.18\times10^{-3}$ W/m², giving $\beta = 10\log_{10}(3.18\times10^{-3}/10^{-12}) = 10\log_{10}(3.18\times10^{9}) \approx 95$ dB.

Everyone has experienced the Doppler effect: when an ambulance approaches, the siren sounds higher-pitched (higher frequency); as it moves away, the pitch drops (lower frequency). This frequency shift occurs because relative motion between source and observer changes how many wavefronts reach the observer per unit time.

General formula:

$f' = f_0\,\dfrac{v \pm v_o}{v \mp v_s}$

where:

• $f_0$ = frequency emitted by the source (Hz)
• $f'$ = frequency perceived by the observer (Hz)
• $v$ = speed of sound in the medium (m/s)
• $v_o$ = velocity of the observer (m/s)
• $v_s$ = velocity of the source (m/s)

Sign convention: Velocities are positive in the direction of approach:

• Numerator ($v \pm v_o$): $+v_o$ if observer moves toward source (increases perceived frequency); $-v_o$ if moving away.
• Denominator ($v \mp v_s$): $-v_s$ if source moves toward observer (denominator decreases, $f'$ increases); $+v_s$ if moving away.

Why does it happen? — Two different physical cases:

1. Moving source ($v_s > 0$, $v_o = 0$): The source "chases" the waves it emits in the direction of motion. Wavefronts bunch up ahead (wavelength $\lambda' = (v - v_s)/f_0$) and stretch behind. The observer ahead hears $f' = v/\lambda' = f_0\,v/(v - v_s) > f_0$.

2. Moving observer ($v_o > 0$, $v_s = 0$): The observer "meets" wavefronts more quickly. Approaching, the relative wave speed is $v + v_o$, so $f' = (v+v_o)/\lambda = f_0\,(v+v_o)/v > f_0$.

The two cases give different formulas but agree in the final result. The general formula unifies them.

Shock waves — supersonic speed ($v_s > v$): When the source exceeds the speed of sound ($v_s > v$), it "outruns" its own waves. Wavefronts accumulate along a cone called the Mach cone. When this cone reaches the observer, a sonic boom is heard — a sudden pressure change. The cone angle is $\sin\theta = v/v_s$. Supersonic jets and bullets produce sonic booms.

Applications:

• Speed radar: an electromagnetic wave (following the same Doppler formula, relativistically) is reflected by a moving vehicle. The frequency shift reveals its speed.
• Doppler ultrasound: ultrasonic waves reflected by moving red blood cells allow measurement of blood flow velocity.
• Astronomical redshift: light from distant galaxies is shifted toward the red (lower frequency) because the universe is expanding. This is the relativistic Doppler effect for light, which allowed Hubble to discover the expansion of the universe.