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Lamb waves

Lamb waves is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Lamb waves rather than just read about it. In short: In solid mechanics, Lamb waves propagate in solid plates or spheres. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the direction perpendicular to the plate.

Lamb waves — main illustration
Lamb waves — illustration

Key takeaways

  • Lamb waves belongs to physics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Lamb waves to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Lamb waves from memory before moving on to harder problems.

Reference excerpt

In solid mechanics, Lamb waves propagate in solid plates or spheres. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the direction perpendicular to the plate. In 1917, the English mathematician Horace Lamb published his classic analysis and description of acoustic waves of this type. An infinite medium supports just two wave modes, longitudinal and shear, traveling at unique velocities; but plates support two infinite sets of Lamb wave modes, symmetric and antisymmetric, whose velocities depend on the relationship between wavelength and plate thickness. Since the 1990s, the understanding and utilization of Lamb waves have advanced greatly, thanks to the rapid increase in the availability of computing power. Lamb's theoretical formulations have found substantial practical application, especially in the field of non-destructive testing. The term Rayleigh–Lamb waves embraces the Rayleigh wave, a type of wave that propagates along a single surface. Both Rayleigh and Lamb waves are constrained by the elastic properties of the surface(s) that guide them.

Lamb's characteristic equations In general, elastic waves in solid materials are guided by the boundaries of the media in which they propagate. An approach to guided wave propagation, widely used in physical acoustics, is to seek sinusoidal solutions to the wave equation for linear elastic waves subject to boundary conditions representing the structural geometry. This is a classic eigenvalue problem. Waves in plates were among the first guided waves to be analyzed in this way. The analysis was developed and published in 1917 by Horace Lamb, a leader in the mathematical physics of his day. Lamb's equations were derived by setting up formalism for a solid plate having infinite extent in the x and y directions, and thickness d in the z direction. Sinusoidal solutions to the wave equation were postulated, having x- and z-displacements of the form

ξ = A x f x ( z ) e i ( ω t − k x ) ( 1 ) {\displaystyle \xi =A_{x}f_{x}(z)e^{i(\omega t-kx)}\quad \quad (1)}

ζ = A z f z ( z ) e i ( ω t − k x ) ( 2 ) {\displaystyle \zeta =A_{z}f_{z}(z)e^{i(\omega t-kx)}\quad \quad (2)}

This form represents sinusoidal waves propagating in the x direction with wavelength 2π/k and frequency ω/2π. Displacement is a function of x, z, t only; there is no displacement in the y direction and no variation of any physical quantities in the y direction. The physical boundary condition for the free surfaces of the plate is that the component of stress in the z direction at z = ±d/2 is zero. Applying these two conditions to the above-formalized solutions to the wave equation, a pair of characteristic equations can be found. These are:

tanh ⁡ ( β d / 2 ) tanh ⁡ ( α d / 2 ) = 4 α β k 2 ( k 2 + β 2 ) 2 ( 3 ) {\displaystyle {\frac {\tanh(\beta d/2)}{\tanh(\alpha d/2)}}={\frac {4\alpha \beta k^{2}}{(k^{2}+\beta ^{2})^{2}}}\ \quad \quad \quad \quad (3)}

for symmetric modes and

tanh ⁡ ( β d / 2 ) tanh ⁡ ( α d / 2 ) = ( k 2 + β 2 ) 2 4 α β k 2 ( 4 ) {\displaystyle {\frac {\tanh(\beta d/2)}{\tanh(\alpha d/2)}}={\frac {(k^{2}+\beta ^{2})^{2}}{4\alpha \beta k^{2}}}\ \quad \quad \quad \quad (4)}

for antisymmetric modes, where

… excerpt ends here. Continue reading the full article.

Illustrations

Lamb waves: Figure 1: Upper and lower, respectively:Extensional (S0) mode with d / λ = 0.6 {\displaystyle d/\lambda =0.6} .Flexural (A0) mode with d / λ = 0.3 {\displaystyle d/\lambda =0.3} .(This is a simplified graphic. It is based on the z component of motion only, so it does not render the distortion of the plate precisely.)
Figure 1: Upper and lower, respectively:Extensional (S0) mode with d / λ = 0.6 {\displaystyle d/\lambda =0.6} .Flexural (A0) mode with d / λ = 0.3 {\displaystyle d/\lambda =0.3} .(This is a simplified graphic. It is based on the z component of motion only, so it does not render the distortion of the plate precisely.)
Lamb waves: Dispersions curves of free Lamb waves for two different Poisson's ratios σ {\displaystyle \sigma } . The x-axis shows the product of angular frequency ω {\displaystyle \omega } and plate thickness d {\displaystyle d} normalized by the shear wave velocity v s {\displaystyle v_{s}} . The y-axis shows the phase velocity v {\displaystyle v} of the Lamb wave normalized by the shear wave velocity. For high frequencies S 0 {\displaystyle S_{0}} and A 0 {\displaystyle A_{0}} modes have the Rayleigh wave velocity, approximate 92 % of the shear wave velocity.
Dispersions curves of free Lamb waves for two different Poisson's ratios σ {\displaystyle \sigma } . The x-axis shows the product of angular frequency ω {\displaystyle \omega } and plate thickness d {\displaystyle d} normalized by the shear wave velocity v s {\displaystyle v_{s}} . The y-axis shows the phase velocity v {\displaystyle v} of the Lamb wave normalized by the shear wave velocity. For high frequencies S 0 {\displaystyle S_{0}} and A 0 {\displaystyle A_{0}} modes have the Rayleigh wave velocity, approximate 92 % of the shear wave velocity.

Worked examples

Example 1 — a first encounter with Lamb waves

Start with the simplest possible case. Write down what Lamb waves claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Lamb waves before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Lamb waves ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Lamb waves

In research
Lamb waves appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Lamb waves in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Lamb waves is common in secondary-school and first-year university syllabi. It links to neighbouring topics Acoustics, Elasticity (physics), Nondestructive testing, so understanding it makes those chapters shorter.
In everyday life
Look for Lamb waves outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Lamb waves in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Lamb waves means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Lamb waves out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lamb waves in simple terms?

In solid mechanics, Lamb waves propagate in solid plates or spheres. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the direction perpendicular to the plate.

Why does Lamb waves matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Lamb waves?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Lamb waves.

Tags

  • Acoustics
  • Elasticity (physics)
  • Nondestructive testing
  • Wave mechanics

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