PHYS2150 Chap.9 Wave, Heat and Laplace Equations
Wave, Heat and Laplace Equations
Three equations encode different physics
Partial differential equations describe fields with several independent variables. The one-dimensional wave equation models propagating or vibrating displacement and contains a second time derivative. The heat equation models diffusion and contains a first time derivative. Laplace’s equation contains no time and describes equilibrium fields.
This structural difference predicts behaviour before calculation: ideal wave modes oscillate, nonconstant heat modes decay, and Laplace solutions are time independent. Initial data specify a starting field and, for the wave equation, its starting velocity. Boundary data prescribe values or normal derivatives at the spatial edge.
A solution must satisfy the equation, every boundary condition and all required initial data.
Separation creates linked ordinary equations
Assume a nonzero product such as u(x,t)=X(x)T(t), substitute it into the PDE, and divide so an x-only expression equals a t-only expression. Because x and t vary independently, both expressions equal one separation constant.
Sign conventions differ between derivations, so solve the ordinary equations actually written instead of memorizing which named sign yields trigonometric functions. Consider positive, zero and negative cases. Boundary conditions eliminate inadmissible families and permit discrete eigenvalues.
A case should be rejected only when the complete boundary system forces all coefficients to zero; otherwise it may contain an essential mode.
Boundary conditions choose the spatial vocabulary
Dirichlet conditions prescribe the field value, while Neumann conditions prescribe the normal derivative or flux. Homogeneous fixed endpoints commonly select sine modes.
Homogeneous insulated endpoints select cosine modes and may retain a constant zero mode. That constant represents the conserved spatial mean in an insulated heat problem and does not decay. Mixed conditions produce a different discrete spectrum. If endpoint values are nonzero, first find a steady function satisfying them and subtract it, leaving a residual problem with homogeneous boundaries.
This shift makes the eigenfunctions usable without changing the underlying physical boundary values.
Initial data choose modal coefficients
After the boundary eigenproblem is solved, superpose the permitted modes. Orthogonality isolates coefficients: multiply the initial profile by one eigenfunction, integrate over the interval, and divide by that mode’s norm.
Treat a constant Neumann mode separately because its normalization differs from the nonzero cosine modes. In the heat equation, a mode with wave number k_n is multiplied by exp(-κk_n²t), so fine-scale structure disappears faster. In the wave equation, each spatial mode receives sine and cosine time factors; initial displacement determines one set of coefficients and initial velocity determines the other.
If the initial profile already contains only a few eigenfunctions, identify them directly.
Verification combines algebra and physics
Check one representative separated mode in the PDE and boundary conditions, then invoke linearity for an appropriate convergent superposition. At the initial time, the series must reproduce the supplied profile and any initial velocity.
Heat solutions should not develop growing high-frequency modes when diffusivity is positive; insulated heat should preserve its mean; ideal wave modes should retain their characteristic frequencies. For Laplace problems on rectangles, select trigonometric and hyperbolic factors that meet the homogeneous edges, then expand the remaining boundary values.
Dimensions provide another audit: diffusivity times time times wave number squared is dimensionless, and wave speed times wave number has frequency units. These checks expose missing squares, wrong separation signs and discarded zero modes.
What this chapter covers
- 01
PDE types and physical models
- 02
Dirichlet and Neumann conditions
- 03
Separation of variables
- 04
Eigenvalues and eigenmodes
- 05
Initial data and modal series
Worked example · free
An insulated heat mode
- 2Translate insulated boundaries.
- 4Retain the constant and cosine modes.
- 3Apply initial data and verify decay.
Key terms
- Separation Constant
- A constant introduced when expressions depending on independent variables are set equal after a product substitution; its allowed values are determined by the boundary problem.
- Boundary Condition
- A rule imposed at the spatial edge of a domain, prescribing the field itself, its normal derivative or another relation needed to select admissible modes.
- Eigenfunction
- A nonzero spatial function satisfying the separated differential equation and homogeneous boundary conditions for one permitted eigenvalue.
- Dirichlet Condition
- A boundary condition that prescribes the value of the field, such as the displacement of a fixed string end or a maintained temperature.
- Neumann Condition
- A boundary condition that prescribes a normal derivative; a zero Neumann condition commonly represents insulation or zero flux.
Wave, Heat and Laplace Equations FAQ
What is the difference between initial and boundary data?
Boundary data act at the spatial edges throughout the time interval and determine which eigenfunctions are admissible. Initial data describe the field at the starting time and determine the coefficient of each permitted mode. A wave equation normally needs both initial displacement and initial velocity because it is second order in time.
Why can a zero mode matter?
Under insulated heat boundaries, a constant spatial function has zero derivative at both ends and therefore satisfies the boundary conditions. Its eigenvalue is zero, so its time factor does not decay. It represents the conserved mean temperature; discarding it would incorrectly force every solution toward zero instead of its initial average.
How is a separated series checked?
Substitute a representative mode into the PDE and each boundary condition, then use linearity for the modal sum. At the initial time, verify the supplied profile and any initial velocity through the coefficient expansion. Finally compare signs and long-time behaviour with the physics: heat modes decay, ideal wave modes oscillate, and Laplace fields have no time dependence.
Exam move
Use the same ordered template for every separation problem. First classify the PDE and translate each verbal boundary statement into a value or derivative equation. Substitute a product, state the sign convention for the separation constant, and solve positive, zero and negative cases in a compact table. Apply both boundary conditions before looking at initial data.
List the permitted eigenvalues, eigenfunctions and normalizations, including any zero mode. Only then project the initial profile and, for waves, the initial velocity. Finish by checking one mode in the PDE, all boundaries, the initial data, dimensions and the expected long-time physics. This order prevents the initial function from dictating an inadmissible spatial family.
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