Normal Modes in Plates and Higher Dimensions
For separable small transverse oscillations of a rectangular membrane or plate idealized by the wave equation, normal modes have spatial factors X(x)Y(y) chosen to satisfy all edge conditions, with squared modal frequency proportional to kx squared plus ky squared.
Why this shows up in the exam
Testing allowed square-plate waveforms · Counting nodal lines · Comparing degenerate mode pairs
Learn the idea
A multidimensional standing pattern must satisfy every boundary independently. On a rectangular plate, each direction contributes its own node pattern; a valid mode is a product of spatial factors that vanish on every clamped edge.
🧠 Memory hook: Every fixed edge must make the shape factor vanish.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- u_mn = A sin(m pi x/Lx) sin(n pi y/Ly) cos(omega_mn t+phi) — fixed-edge rectangular mode shape
- k_mn² = (m pi/Lx)² + (n pi/Ly)² — combined spatial wavenumber
How to approach it
- 1Apply each edge condition to the candidate
- 2Read integer mode numbers in both directions
- 3Combine orthogonal wavenumbers through squares
Common slip-ups that cost marks
- •Checking only one pair of edges
- •Adding directional wavenumbers instead of their squares
- •Treating a two-dimensional mode as a travelling plane wave
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
More from Oscillations and Waves
Superposition, Beats, and Resonance
Explore the principles of superposition, formation of beats, beat frequency, resonance, and phase difference in wave phenomena.
Standing Waves and Harmonics
Understand the formation of standing waves, harmonics in strings and organ pipes (open and closed), and the relationship between displacement and pressure in stationary waves.
Velocity and Acceleration in SHM
Explore how velocity and acceleration vary with displacement and time in SHM, including maximum values and their physical significance.
Simple Pendulum and Spring Systems
Study the time period formulas, energy considerations, and behavior of simple pendulums and spring-mass systems, including effects of non-inertial frames and spring combinations.
Basics of Simple Harmonic Motion (SHM)
Understand the defining features of SHM, including its standard equations, conditions, and the relationship between displacement, velocity, and acceleration.
Wave Motion and Wave Equation
Learn the fundamentals of wave motion, including the wave equation, its parameters, direction, and the distinction between particle and wave velocities.