MECH7029 Chap.4 Wave Equation and Acoustic State Relations
Wave Equation and Acoustic State Relations
Propagation emerges from three balances
The homogeneous wave equation is derived from linearised mass conservation, momentum conservation and isentropic thermodynamics. The chapter then links pressure, density, temperature, particle velocity and displacement for a travelling wave.
Every relation carries a model boundary: small disturbance, stationary mean medium, propagation direction and a wave field rather than a standing or reactive near field.
Linearisation is a physical decision
Total density, pressure and velocity are divided into steady means and small fluctuations.
Retaining only first-order disturbance terms removes products of fluctuations while preserving the leading wave dynamics. Mass conservation connects density change with velocity convergence, momentum conservation links pressure gradient with acceleration, and isentropic closure ties pressure to density.
Differentiating and eliminating velocity leaves the homogeneous propagation equation.
Each omitted effect points toward later material. A volume input changes the mass balance, an external force changes momentum, heating changes thermodynamic closure, and turbulent momentum flux restores nonlinear stress.
The derivation therefore acts as a source-classification map rather than only a route to one formula.
State relations depend on wave structure
For a single travelling plane wave, pressure and particle velocity are locally proportional through characteristic impedance, with sign set by propagation direction. Velocity divided by angular frequency gives displacement amplitude.
Adding an equal reflected wave produces a standing pattern in which local pressure and velocity maxima separate, so the travelling-wave shortcut no longer describes the total field. Sound speed is set by elastic response divided by inertia and varies with temperature even when acoustic amplitude is unchanged.
What this chapter covers
- 01
Ideal-gas and isentropic relations
- 02
Linearised mass conservation
- 03
Linearised momentum conservation
- 04
One-dimensional wave equation
- 05
Travelling-wave pressure and velocity
- 06
Sound speed and particle displacement
Worked example · free
Recover particle motion from pressure
- 4Use characteristic impedance to find peak particle velocity.
- 2Convert frequency to angular frequency.
- 2Find peak displacement and interpret its scale.
Key terms
- Linearisation
- Retention of first-order fluctuations while higher-order products are neglected.
- Isentropic Process
- A reversible adiabatic change with constant entropy.
- Lagrangian Derivative
- The rate of change following a moving fluid particle.
- Wave Equation
- The propagation relation obtained after velocity is eliminated from linear balances.
- Characteristic Impedance
- The travelling-wave pressure-to-velocity ratio in a uniform medium.
Wave Equation and Acoustic State Relations FAQ
Why does temperature change in an adiabatic sound wave?
Adiabatic means no external heat transfer, not constant temperature. Mechanical compression raises molecular energy and temperature, while expansion lowers them. The reversible changes of pressure, density and temperature are linked through the isentropic relation.
Where does sound speed appear in the derivation?
The thermodynamic closure relates pressure and density fluctuations, and its slope is sound speed squared. Substituting that relation after mass and momentum equations are combined sets the coefficient connecting time and space derivatives.
When is pressure equal to density times sound speed times velocity?
The relation applies to a single travelling plane wave in a uniform medium, with sign determined by propagation direction. A superposition of opposing waves can have pressure maxima where velocity vanishes, so the total field needs separate amplitudes.
Which assumptions should accompany the homogeneous wave equation?
State that disturbances are small, the reference medium is uniform and stationary, and the pressure-density closure is isentropic. Boundary and initial conditions are still needed to select a particular field. If mean flow, strong nonlinearity, external force, heat release or changing volume is important, the source-free equation must be extended rather than applied silently.
Exam move
Rebuild the derivation first as a causal diagram and then as algebra. Write the perturbation split, linear mass balance, linear momentum balance, isentropic closure and elimination step in words. Beside every step, note the discarded term and the physical situation that could make it important. Only after that should you reproduce the derivatives.
Use dimensional checks to confirm that both sides of the wave equation have pressure per length squared. For state calculations, create a column for peak or RMS convention and another for propagation direction. Starting with one pressure amplitude, obtain particle velocity from characteristic impedance and displacement from angular frequency.
Repeat after reversing direction so the velocity sign changes while pressure amplitude does not. Next add a reflected wave and sketch pressure and velocity nodes; write explicitly why the single-wave impedance relation is no longer a pointwise description of the total field. Include one temperature change problem for sound speed and predict the direction before substituting.
Finish revision by classifying four broken assumptions as mass, momentum, thermodynamic or nonlinear-stress sources. The goal is to see every formula as the consequence of a model, not as an isolated identity. Record whether each error came from linearisation, sign, amplitude convention, angular frequency or misuse of the travelling-wave condition.
End with limiting-case checks at zero reflection and equal incident and reflected amplitude; their contrast exposes whether a relation belongs to a travelling or standing field.
