GRE Physics Companion: Helmholtz Equation, Boundary-Value Problems, and Eigenfunction Expansions

GRE Physics Companion: Helmholtz Equation, Boundary-Value Problems, and Eigenfunction Expansions

The key spatial equation is

(1)

or

(2)

For waves,

(3)

1 High-value GRE facts

1. harmonic time dependence gives the Helmholtz equation.
2. Dirichlet conditions specify field value.
3. Neumann conditions specify Normal derivative.
4. Homogeneous Dirichlet Laplacian eigenvalues are positive.
5. Homogeneous Neumann problems can contain a zero mode.
6. Distinct eigenvalues give orthogonal eigenfunctions for self-adjoint boundary conditions.
7. Rectangular Dirichlet eigenfunctions are products of sine functions.
8. Circular Dirichlet eigenfunctions involve Bessel functions.
9. Expansion coefficients are obtained by projection.
10. Wave frequencies satisfy ω_n = c.

Part I: Original GRE-style problems

Problem 1: Helmholtz equation

If

(4)

then the wave equation gives

1. ∇²Φ − k²Φ = 0
2. ∇²Φ + k²Φ = 0
3. ∇Φ + kΦ = 0
4. Φ_tt + k²Φ = 0
5. ∇²Φ = 0 for all ω

Problem 2: Dirichlet condition

A homogeneous Dirichlet condition is

1. ∂Φ∕∂n = 0
2. Φ = 0
3. ∇²Φ = 0
4. Φ_t = 0
5. Φ = constant everywhere

Problem 3: Neumann condition

A homogeneous Neumann condition is

1. Φ = 0
2. ∂Φ∕∂n = 0
3. ∇Φ = 0 everywhere
4. Φ_tt = 0
5. k = 0

Problem 4: fixed interval eigenvalues

On 0 < x < L with Φ(0) = Φ(L) = 0, the eigenvalues are

1. λ_n = nπ∕L
2. λ_n = (nπ∕L)²
3. λ_n = n∕L²
4. λ_n = 2πn∕L
5. λ_n = 0 for all n

Problem 5: Neumann zero mode

For homogeneous Neumann conditions on a closed interval, the n = 0 mode is

1. forbidden
2. a constant eigenfunction with λ = 0
3. a sine function
4. always singular
5. a mode with infinite frequency

Problem 6: orthogonality

If λ_m≠λ_n for a self-adjoint Laplacian problem, then

1. Φ_m = Φ_n
2. ∫ Φ_mΦ_n dV = 0
3. λ_m + λ_n = 0
4. both eigenfunctions vanish everywhere
5. both eigenfunctions are constant

Problem 7: square degeneracy

For a square Dirichlet membrane, the pair (1, 2) and (2, 1)

1. has different eigenvalues
2. is degenerate
3. contains a zero mode
4. exists only for Neumann boundaries
5. is forbidden

Problem 8: projection coefficient

For an orthogonal basis, the coefficient of Φ_n in f is

1. ⟨f, Φ_n⟩
2. ⟨f, Φ_n⟩∕⟨Φ_n, Φ_n⟩
3. ⟨Φ_n, Φ_n⟩∕⟨f, Φ_n⟩
4. λ_nf
5. ω_nf

Problem 9: wave frequency

If a spatial eigenvalue is λ_n, the corresponding wave angular frequency is

1. λ_n∕c
2. cλ_n
3. c
4. 
5. 1∕λ_n

Problem 10: rectangular eigenvalue

For a rectangle with Dirichlet boundaries,

(5)

is

1. π(m∕a + n∕b)
2. π²(m²∕a² + n²∕b²)
3. (m + n)²
4. m²n²∕(ab)
5. independent of geometry

Problem 11: circular geometry

For a circular Dirichlet domain, radial eigenvalues are selected by

1. sin(ka) = 0 for every angular order
2. J_m(ka) = 0
3. Y _m(ka) = 0 only
4. ka = m + n
5. k = 0 only

Problem 12: boundary conditions

Changing boundary conditions while keeping the same differential operator generally

1. leaves the spectrum unchanged
2. changes the allowed eigenfunctions and eigenvalues
3. changes the spatial dimension
4. removes orthogonality in every case
5. forces k = 0

Part II: Complete worked solutions

Solution 1

Substitution gives

(6)

With k = ω∕c,

(7)

Answer: (B).

Solution 2

Homogeneous Dirichlet means the field itself vanishes on the boundary. Answer: (B).

Solution 3

Homogeneous Neumann means the outward normal derivative vanishes. Answer: (B).

Solution 4

The fixed interval gives

(8)

so

(9)

Answer: (B).

Solution 5

The constant function satisfies the Neumann boundary condition and has zero Laplacian. Answer: (B).

Solution 6

Self-adjointness gives orthogonality for distinct eigenvalues:

(10)

Answer: (B).

Solution 7

On a square,

(11)

Therefore (1, 2) and (2, 1) have the same eigenvalue. Answer: (B).

Solution 8

Projection onto an orthogonal but not necessarily normalized basis gives

(12)

Answer: (B).

Solution 9

Since

(13)

and

(14)

(15)

Answer: (C).

Solution 10

Separation in a rectangle gives

(16)

Answer: (B).

Solution 11

regular circular eigenfunctions use J_m(kr), and the fixed rim requires J_m(ka) = 0. Answer: (B).

Solution 12

The boundary conditions are part of the eigenvalue problem and therefore change the allowed spectrum. Answer: (B).

2 GRE checklist

1. Convert wave problems to Helmholtz form only after assuming harmonic time dependence.
2. Distinguish field value from normal derivative at a boundary.
3. Keep track of the sign convention in −∇²Φ = λΦ.
4. Remember the Neumann zero mode.
5. Use self-adjointness to justify orthogonality.
6. Use projection divided by mode norm for expansion coefficients.
7. Convert spatial eigenvalues to wave frequencies with ω = c.

References

References