Appendix C: Derivation of the 1D Wave Equation

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<ul><li> 1. Appendix CDerivation of 1-D wave equationIn this appendix the one-dimensional wave equation for an acoustic medium is derived,starting from the conservation of mass and conservation of momentum (Newtons SecondLaw).DerivationHere we will derive the wave equation for homogeneous media, using the conservation ofmomentum (Newtons second law) and the conservation of mass. In this derivation, wewill follow (Berkhout 1984: appendix C), where we consider a single cube of mass whenit is subdued to a seismic disturbance (see gure (C.1)). Such a cube has a volume Vwith sides x, y and z. Conservation of mass gives us: m(t0 ) = m(t0 + dt) (C.1)where m is the mass of the volume V, and t denotes time. Using the density , theconservation of mass can be written as: (t0 )V (t0 ) = (t0 + dt)V (t0 + dt) (C.2)Making this explicit: 0 V = (0 + d)(V + dV ) = 0 V + 0 dV + V d + ddV (C.3)Ignoring lower-order terms, i.e., ddV, it follows that d dV = (C.4) 0 V 128 </li> <li> 2. p+pz z p+p y y x p+p x Figure C.1: A cube of mass, used for derivation of the wave equation.We want to derive an equation with the pressure in it so we assume there is a linearrelation between the pressure p and the density: K dp = d (C.5) 0where K is called the bulk modulus. Then, we can rewrite the above equation as: dV dp = K (C.6) Vwhich formulates Hookes law. It shows that for a constant mass the pressure is linearlyrelated to the relative volume change. Now we assume that the volume change is only inone direction (1-Dimensional). Then we have: dV (x + dx)yz xyz = V xyz dx = (C.7) xSince dx is the dierence between the displacements u x at the sides, we can write: dx = (dux )x+x (dux )x (dux ) (vx ) = x = dt x (C.8) x x 129 </li> <li> 3. where vx denotes the particle velocity in the xdirection. Substitute this in Hookes law(equation C.6): vx dp = K dt (C.9) xor 1 dp vx = (C.10) K dt xThe term on the left-hand side can be written as : 1 dp 1 p p = + vx (C.11) K dt K t xIgnoring the second term in brackets (low-velocity approximation), we obtain for equation(C.10): 1 p vx = (C.12) K t xThis is one basic relation needed for the derivation of the wave equation. The other relation is obtained via Newtons law applied to the volume V in thedirection x, since we consider 1-Dimensional motion: dvx Fx = m (C.13) dtwhere F is the force working on the element V. Consider the force in the xdirection: Fx = px Sx p p = x + dt Sx x t p V (C.14) xignoring the term with dt since it is small, and S x is the surface in the xdirection, thusyz. Substituting in Newtons law (equation C.13), we obtain: p dvx V = m x dt dvx = V (C.15) dt 130 </li> <li> 4. We can write dvx /dt as vx /t; for this we use again the low-velocity approximation: dvx vx vx vx = + vx (C.16) dt t x tWe divide by V to give: p vx = (C.17) x tThis equation is called the equation of motion. We are now going to combine the conservation of mass and the equation of motion.Therefore we let the operator (/x) work on the equation of motion: p vx = x x x t vx = (C.18) t xfor constant . Substituting the result of the conservation of mass gives: 2p 1 p = (C.19) x2 t K tRewriting gives us the 1-Dimensional wave equation: p2 2p 2 =0 (C.20) x K t2or 2p 1 2p 2 2 =0 (C.21) x2 c tin which c can be seen as the velocity of sound, for which we have: c = K/. 131 </li> </ul>