Math 2.0

Mechanics of deformable solids Definition

Stoneley wave

Builds on ElastodynamicsS-waveRayleigh wave

Definition

Let two homogeneous isotropic linear elastic solids fill the half-spaces x3<0x_3 < 0 and x3>0x_3 > 0. Let them be in welded contact along the plane x3=0x_3 = 0, so that displacement and traction are continuous there. Let cS,1c_{S,1} and cS,2c_{S,2} be their S-wave speeds. A Stoneley wave is a nonzero elastodynamic solution without body force of the form u(x,t)=Re⁡[(U1(x3), 0, U3(x3))eik(x1−ct)],k>0,u(x, t) = \operatorname{Re} \left[ \left( U_1(x_3), \, 0, \, U_3(x_3) \right) e^{i k (x_1 - c t)} \right], \qquad k > 0, whose amplitude tends to 00 as x3→−∞x_3 \to -\infty and as x3→+∞x_3 \to +\infty.

Remarks

Decay on both sides forces c<min⁡(cS,1,cS,2)c < \min(c_{S,1}, c_{S,2}), since otherwise one of the shear parts would oscillate instead of decay away from the interface. The four interface conditions give a 4×44 \times 4 linear system for the amplitudes, and a Stoneley wave exists when its determinant vanishes for some such cc. As for the Rayleigh wave, cc does not depend on kk.

Existence holds only for a restricted range of the ratios of the densities and elastic constants of the two solids. Non-example: when the two solids have the same density and the same elastic constants, the body is one homogeneous space, and no wave tied to the plane x3=0x_3 = 0 exists.

Another non-example: a Rayleigh wave on the free surface of a single half-space decays only on one side; it has no second medium, so it is not a Stoneley wave.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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  1. Elastodynamics concept
  2. Rayleigh wave concept
  3. S-wave concept
  4. Stoneley wave definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Stoneley_wave
About this page
Tag92NBHACRFV
URIm2://92NBHACRFV
Kinddefinition
WritingW0 stub
MSC74J15, 86A15
DefinesStoneley wave
UsesElastodynamics, S-wave, Rayleigh wave
Changed3 November 2026

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