Chapter 5 Sound Waves

Chapter 5

Sound Waves

5.1 Preliminaries

Sound waves exist in solids, liquids and gases. To allow for propagation of sound waves, themedium must be compressible.

An incompressible fluid behaves like a rigid body. The body moves without deformation andoscillations on one side are transmitted instantaneously to the other (arbitrarily distant) side:

rigid body

instantaneous propagation

this corresponds to an infinite sound speed!

A compressible fluid behaves like an elastic solid. Now oscillations propagate through the solidin form of compression waves. Their speed is finite and depends on the elastic properties, pres-sure, temperature etc.

elastic body

finite velocity

71

72 CHAPTER 5. SOUND WAVES

If the amplitude of these compression waves is (infinitesimally) small, they are called “acousticwaves” or “sound waves”.

5.2 Sound speed

We consider a front that separates two regions with different pressure and density. The frontmoves to the left with constant velocity c. The fluid left from the front is in rest.

moving front

c

p

u=u + du

p=p + dpρ

xdρ=ρ + ρ

0

0

0

0

0

u =00

In the co-moving frame (with c to the left) one has a stationary front.

x

0

stationary front

u= c + duu = c

Now we take a finite volume around the front with surface A

A

c c + du

5.3. WAVE EQUATION FOR SOUND WAVES 73

Conservation of mass yields

c ρ0 A c du ρ0 dρ A c ρ0 A ρ0 A du c A dρ

or

du cρ0

dρ (5.1)

If dρ 0 (compression wave), the fluid behind the front moves with du 0 (in the direction ofthe front motion).

Now we take the conservation of the momentum (no friction, Euler eq.):

p0 A p0 dp A net force on V

A ρ0 c c du A ρ0 c c change of x-momentum

or

dp ρ0 c du (5.2)

Eliminating du from (5.1) and (5.2) one finds the important result

c2 dpdρ

To compute the speed of sound one needs a state equation p pρ .

5.3 Wave equation for sound waves

For small amplitude waves, viscosity and nonlinearities can be neglected. The linearized Eulereq. and continuity eq. read

ρ∂ v∂t

grad p (5.3)

∂ρ∂t

div v ρ (5.4)


5.3.1 Compression waves

We use the decomposition

ρ v ρ v1 ρ v2

withcurlρ v1 0 ρ v1 gradΦ

anddivρ v2 0 ρ v2 curl A

The first part describes a pure compression without shearing or vortices. The second part corre-sponds to a shearing without volume change.

Inserting this into (5.3) yields

gradΦ curl ˙A grad p (5.5)

andcurl ˙A 0

The vortices remain constant and are conserved. Then we can integrate (5.5) to

Φ p0 p (5.6)

with a certain constant p0. Inserting the decomposition into (5.4) one gets

ρ divgradΦ divcurl 0

A ∆Φ (5.7)

Again we need a state equation of the form p p ρ . Then we can differentiate (5.6) with respectto time and use (5.7)

Φ p dpdρ

ρ dpdρ

∆Φ

or

Φ c2 ∆Φ 0

5.3. WAVE EQUATION FOR SOUND WAVES 75

This is a wave equation for sound waves with phase speed c2 dp dρ. From (5.6) the pressurewaves can be computed.

5.3.2 State equation

To evaluate c, Isaac Newton used the state equation of a perfect gas:

p ρ R T (5.8)

For air at room temperature this gives c ! 290 m/s, compared to c 340 m/s from the experiment.Newton assumed “unclean air” being the reason for the large discrepancy.

About 100 years later, Laplace showed that the compression is not isothermal but adiabatic (orisentrop). The temperature changes during compression periodically. But the motion is so fast,that the temperature fluctuations are not transported to the environment by heat flux.

For an adiabatic process, the relation between pressure and density reads

p const " ργ

where γ cp cV is the adiabatic exponent and cp, cV is the specific heat under constant pressureand volume, respectively. For a mono-atomic perfect gas one has γ 5 3, for a di-atomic gasγ 7 5.

Thusdpdρ

γ " const " ργ # 1

Using (5.8) one determines the constant to RT ρ1 # γ and finally finds

c %$ γ R T &a value, which is in excellent agreement with the experiment.

Chapter 6

Surface Waves

6.1 Preliminaries

We consider waves on the surface of a liquid layer (river, lake, ocean)

P0: external pressureρ: density of fluid

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x

yρ

z( )

0p h x, y, t

* find equation for h + x , y , t -* find the internal motion of the fluid .v* can instabilities occur ? / (Part III)

Assumptions and approximations:

* viscosity is not important / Euler equations* no vorticity, curl .v 0 0 / Potential flow* incompressible fluid, div .v 0 0

77

78 CHAPTER 6. SURFACE WAVES

1 small surface deflection, 2 h 3 h0 2h0 4 1

6.2 Gravity waves

6.2.1 equations for flow

Euler equations 5 ρ 6 const)

˙7v 895 7v : ∇ ; 7v 63 1ρ

grad 5 P 8 U ;with (Potential flow) 7

v 6 ∇Φ

and the formula 5 7v : ∇ ; 7v 6 12

∇v2<>=?>@∇ A ∇Φ B 2 3 7v CD5 ∇ C 7v ;< =? @E 0

We find

∇ F Φ 8 125 ∇Φ ; 2 G 63 ∇ F P 8 U

ρG (6.1)

or, after integration

Φ 6 3 P 8 Uρ

3 5 ∇Φ ; 22

∆Φ 6 0 (6.2)

This are the basic equation for an incompressible, vortex free fluid (cmp. part I, chapt. 3.4)

The water in a constant gravitation field has the potential energy:

U 6 ρgz 8 U0

6.2.2 Equation for the location of the free surface

Let the surface be located at z 6 h 5 x H y H t ;

6.2. GRAVITY WAVES 79

x

y

z

h (x, y, t), 2D − plane

If there is a vertical velocity component, the surface moves with that velocity: h I vz J z I h Kh

v (h)z

but also a horizontal velocity takes the surface with it, according to:

h IL vx J z I h K ∂xh

vx (h)

both together yields (in three dimensions)

h IL vx M hN>OP>Q∂xΦ

∂xh L vy M hN>OP>Q∂yΦ

∂yh R vz M hN>OP>Q∂zΦ

(6.3)

or, using the potential:

h IL ∇2Φ M h S ∇2h R ∂zΦ M h (6.4)

Now we evaluate eq. (6.2) at the surface z I h:

Φ M h IL g J h L h0 KN OP QU T z U h V L 1

2 J ∇Φ K 2h (6.5)

where we usedU I ρgz R U0 W with U0 I P0 L ρgh0


6.2.3 Basic equations and linear solutions

∆Φ X 0 (6.6)

Φh X Y g Z h Y h0 [ Y 12Z ∇Φ [ 2h (6.7)

h X Y ∇Φ \ h ] ∇h ^ ∂zΦ \ h (6.8)

Now we assume that the basic state is that of a flat surface h X h0 where the fluid is in rest, Φ X 0(hydrostatic solution)

Consider small deviations from that state

η Z x _ t [ X h Z x _ t [ Y h0 _ Φ Z x _ t [ (6.9)

eqs. (6.7), (6.8) can be linearized:

Φ \ z X Y gηη X ∂zΦ \ h ` Xba ∂zΦ ^ 1

gΦ X 0 (6.10)

We assume a solution in form of waves:

Φ X ξ Z t [ ] f Z z [ eikx (6.11)

inserting this into (6.6) yields

f cdceY k2 f X 0 f f Z z [bg e hji k ilk z m h0 nand with the boundary condition (infinitely deep layer)

Φ Z z foY ∞ [ X 0 f f Z z [bg e i k i k z m h0 n (6.12)

Substitute (6.11), (6.12) into (6.10) gives

\ k \ ζ ^ 1g

ζ X 0 f ζ Z t [ X e h iωt _ (6.13)

6.2. GRAVITY WAVES 81

the equation of an harmonic oscillator with the frequency

ω p%q r k r gThus we have as a solution of the linearized problem

Φ p Ae s k s t z u h0 v cos w kx x ωt y (6.14)

h p h0 x Akg

sin w kx x ωt y (6.15)

and from there the velocity components

vx p ∂xΦ pz kAe s k s t z u h0 v sin w kx x ωt y (6.16)

vz p ∂zΦ pr k rAe s k s t z u h0 v cos w kx x ωt y (6.17)

This corresponds to traveling waves with the phase velocity

c p ωk

The dispersion relation readsω p q kg

Using this, the phase velocity can be expressed as

c p| gkp | gλ

2π

– the longer the wave length, the faster the wave propagates

How do the trajectories of volume element (its path) look?

To answer this, one has to solve the system

dxdt

p vx pz kAe s k s t z u h0 v sin w kx x ωt y (6.18)

dzdt

p vz pr k rAe s k s t z u h0 v cos w kx x ωt y (6.19)

two coupled nonlinear ODE’s which can be solved only numerically.


Approximation: ~dv ~ c

With the initial condition x0 x t 0 z0 z t 0 one can integrate

x t x0 t0

vx x0 z0 t dt x0 a cos kx0 ωt coskx0 (6.20)

z t z0 t0

vz x0 z0 t dt z0 b sin kx0 ωt sinkx0 (6.21)

with a Akω

e k l z0 h0 (6.22)

b A ~ k ~ω

e k z0 h0 (6.23)

– volume elements travel on circles with radius ~ a ~ ~ b ~ e k z0

– in time average, particles don’t travel at all!

c

But: nonlinear corrections leads to the so-called “Stokes drift”, an average velocity~dvs ~ a2

and parallel to k.

6.3. THE SHALLOW WATER EQUATIONS 83

6.3 The Shallow Water equations

Now we consider surface deformations in form of long waves. This does not mean only harmonicwaves but can be any other form. It is important that the dimension (extension) in horizontaldirection is large compared to the depth of the fluid.

x

z

h

Example of a "long wave"

0

δ h0 1

Examples for “long waves” are:

ocean waves near the shore Tsunamis Waves on a canal

To arrive at a dimensionless formulation of the problem, the variables of eqs. (6.6), (6.7), (6.8)are scaled in the following way:

x x z z h0

h h h0 (6.24)

t t τ Φ Φ 2

τ(6.25)

Then eqs. (6.6), (6.7), (6.8) read

0 ∂2zzΦ δ2∂2

xxΦ (6.26)

˙Φ G h 1 12 ∂xΦ 2 1

2δ2 ∂zΦ 2 (6.27)

δ2 ˙h δ2∂xΦ ∂xh ∂zΦ (6.28)

(from here, we suppress the tildes). The non-dimensional number

G g h0 τ2 2


is called “gravitation number”.

Trick: we solve (6.26) by iteration (systematic perturbation analysis with respect to small δ):

Φ Φ 0 ¡ δ2Φ 2 ¡ δ4Φ 4 ¡£¢¤¢¤¢ (6.29)

this inserted into (6.26) gives:

∂2zzΦ 0 ¡ δ2 ¥ ∂2

zzΦ 2 ¦¡ ∂2xxΦ 0 ¨§©¡ δ4 ¥ ∂3

zzΦ 4 ¦¡ ∂2xxΦ 2 ¨§©¡ª¢¤¢¤¢ 0 (6.30)

Since δ can be arbitrary, terms with the same order of δ must vanish:

« order δ0

∂2zzΦ 0 0 ¬ Φ 0 f1 x ® t ¯ ¡ f2 x ® t ¯° z (6.31)

with the boundary condition on the ground (z 0)

vz z 0 ¯b ∂Φ∂z ±±±± z ² 0

f2 0 ³ f2 0 (6.32)

and finally the important result

Φ 0 Φ 0 x ® t ¯ (6.33)« order δ2:

∂2zzΦ 2 ´ ∂2

xxΦ 0 ´ Φ 0 ¶µ µ (6.34)¬ Φ 2 x ® z ® t ¯· ´ Φ 0 µ µ ° z2

2¡ f3 x ® t ¯¸ ¹º »² 0 (b. c)

° z ¡ f4 x ® t ¯ (6.35)

« order δ4

- in the same way ¢¤¢¤¢We write down the result up to the order δ4:


Φ ¼ x ½ z ½ t ¾·¿ Φ À 0 Á ¼ x ½ t ¾ÃÂ δ2 ÄÆÅ Φ À 0 Á¶Ç ÇÉÈ z2

2Â f4 ¼ x ½ t ¾ÆÊ

Â δ4 Ä Φ À 0 Á¶Ç Ç Ç ÇeÈ z4

24Å f ËdË4 È z2

2Â f6 ¼ x ½ t ¾ÆÊÌÂªÍ¤Í¤Í (6.36)

– we know the z-dependence of Φ explicitly !!

– if Φ À 0 Á ¼ x ½ t ¾ is known, Φ ¼ x ½ z ½ t ¾ can be determined.

Now we insert this into (6.27), (6.28) and take the lowest, non-trivial order:

Φ À 0 Á ¿ Å G ¼ h Å 1 ¾ Å 12 Î ∂xΦ À 0 Á¨Ï 2

(6.37)

h ¿ Å ∂xΦ À 0 ÁÐÈ ∂xh Å h È ∂2xxΦ À 0 Á (6.38)

or in two horizontal dimensions (x ½ y)

Φ ¿ Å G ¼ h Å 1 ¾ Å 12¼ ∇2Φ ¾ 2 (6.39)

h ¿ Å ∇2Φ È ∇2h Å h È ∆2Φ (6.40)

These are the Shallow Water equations.

Advantage: only two equations instead of three

Big advantage: one spatial dimension is eliminated!

3D Ñ 2D

2D Ñ 1D

6.3.1 The linearized Shallow Water equations

We consider small deviations η from the constant depth h0 ¿ 1:


η Ò h Ó 1

Then Φ is also small and (6.37), (6.38) or (6.39), (6.40) can be linearized:

Φ Ò Ó Gη (6.41)

η Ò Ó ∆2Φ (6.42)

Differentiating (6.42) with respect to time and eliminating Φ yields a wave equation for η

η Ó c2∆2η Ò 0

with the phase velocity

c ÒÔ G

rescaling all variables gives the velocity in dimensional form

c ÒÖÕ gh0× phase velocity of long waves is constant!× it depends only on the depth of the layer


6.3.2 Numerical solutions of the nonlinear Shallow Water equations

time evolution in 1D

x

t

x

th

snapshot in 2D


6.3.3 Shallow water waves on a modulated ground

z

x

h(x,t)

f(x)

z=1

H(x) n

boundary condition on the ground:

n ØÚÙv Û n Ø ∇Φ Û 0

Deriving the Shallow Water equations in the same manner as above, this gives rise to two newterms (underlined)

Φ Û Ü G Ý h Ü 1 ÞßÜ 12Ý ∇2Φ Þ 2 (6.43)

h Û ÜàÝ ∇2h ÞßØÝ ∇2Φ ÞßÜ h ∆2Φ á Ý ∇2 f ÞßØeÝ ∇2Φ Þâá f ∆2Φ (6.44)

linearizing again yields a wave equation, now of the form

h Ü G ∇2 ãH Ý x ä y Þ ∇2h åâÛ 0 (6.45)

with H Û 1 Ü f denoting the real depth of the flat water. If we neglect ∇2H (corresponding tosmall changes of the surface on the length scale of the waves), (6.45) describes waves with spacedependent velocity

cp Ý x ä y ÞæÛç G H Ý x ä y Þ (6.46)

It is obvious that waves slow down if they reach a shallower region. In the mean time their wavelength decreases:

λ Û 2πkÛ 2π

ωcp Û 2π

ω è GH é H12 (6.47)


Numerical solution of waves on a beach with constant slope

x

Profil h(x)

f(x)

~ H −1/4

If the ground has a small slope, Green’s law can be derived from weakly non-linear theory (seetextbooks, e.g. Lamb, Hydrodynamics, Cambridge Univ. Press):

A ê H ë 14 (6.48)

From there one sees that the amplitude of waves increases if they approach the shore. We shallreturn to this issue in the sect. on Tsunamis.


6.3.4 Generation of waves by a time-dependent ground

z

x

h(x,t)

z=1

nf(x,t)

H(x,t)

The boundary conditions now have the form:

n ì¤ív î n ì ∇Φ î f

This gives another term (underlined)

Φ î ï G ð h ï 1 ñï 12ð ∇2Φ ñ 2 (6.49)

h î ïàð ∇2h ñòìeð ∇2Φ ñßï h ∆2Φ ó9ð ∇2 f ñßìeð ∇2Φ ñÃó f ∆2Φ ó f (6.50)

If Φ î const (fluid in rest) ô h î f ô h ð t ñõî f ð t ñ ó const

ö the ground motion is equal to the surface motionö no time delay, reason: fluid is assumed to be incompressibleö A ground motion may generate waves:

linearized wave equation with H î 1 ï f :

1G

Φ ï ∇2 ÷H∇2Φ øÃî f îï H (6.51)

This is an inhomogeneous wave equation. It can be formally solved using an appropriateGreen’s function:

Φ ð x ù y ù t ñõîûúüú dx ý dy ýÚú dt ý D ð x ï x ýþù y ï y ýÿù t ï t ýñ H ð x ýÿù y ý ù t ý ñ (6.52)


Example

consider the following localized ground motion:

Hx y t 1 v0 t δ

x δ

y 0 t t0

1 v0 t0 δx δ

y t0 t

(6.53)

0 < t < t 0 t > t 0t = t

0

ground

t = 0

solution: circular waves

t=4t=6

t=8

t=2φ

r

snapshots at various times t


Numerical solution in two dimensions

We chosef x y t a e r2 β2

cosΩt (6.54)

oscillating ground localized at r 0 with a gaussian distribution. slopy ground (ramps in x-direction, minimum in the center).


6.3.5 Tsunamis

The notion “Tsunami” was coined by Japanese fishermen and means “wave in harbor”. Thefishermen went out to the sea during the night for fishing. On their return, they found the harbordestroyed by a flood. Since they didn’t notice anything unusual on the open sea, they thoughtthat these waves were generated in the harbor.

Tsunamis are caused by seaquakes or landslides More than 80 Tsunamis observed in the last 10 years

– Christmas 2004, Sri Lanka, India, Thailand, more than 200 000 victims

– Lissabon 1755, caused by the big earthquake 60 000 victims

– Krakatau 1883, a wave was generated that traveled 7 times round the earth

– Japan, 1896, a Tsunami called “Sanriku” caused waves with amplitudes up to 23meters

What is the difference between a Tsunami and waves generated by wind?

Wind accelerates the fluid on a thin layer at the surface of the sea. Waves generated by wind areshort waves or deep water waves.

Due to the generation of a Tsunami on the ground of the sea, the whole water column over theseismic center is elevated:

ground

surface

seaquake

Thus, the fluid over the whole depth moves. Although the fluid motion is rather slow comparedto that caused by wind waves, its kinetic energy is enormous due to the large mass in motion

Waves generated by a seaquake are long waves or shallow water waves.


c

deep water waves (short waves) shallow water waves (long waves)

On the high seas (off shore) the wave amplitude is very small: 10 - 50 cm The wave length is 100 km The water depth is 4 - 7 km For Tsunamis, the Shallow-Water theory applies

There we found a relation between phase velocity and water depth:

c g h0

If we use g 9 81 m/s2 and h0 4000 m we find

c 200 m/s 700 km/h

A Tsunami may cross an ocean within a few hours! There is almost no damping, because the particle velocity is very slow.

v Ah0

c

with A = 50 cm, h0 = 4000 m, c 200 m/s one gets


v 2 ! 5 cm/s

This cannot be measured on the surface, because it is completely covered by the natural motion(wind). Tsunamis can only be detected well below the surface, where the water is usually notmoving (or only in large scaled streams).

For the frequency, we can estimate

ν " ω2π" c # k

2π" c

λ

Taking λ " 100 km and c " 200 m/s one has ν 0 $ 002 Hz, corresponding to ∆t 500 s betweentwo consecutive waves.

From Green’s law we know that the amplitude increases by approaching the shore:

A % H & 14

The water velocity is also a function of the depth:

v " AH

c

Since c % H12 we finally have a rather strong increase of the water velocity while a Tsunami

reaches the shores:

vv0"(' H0

H ) 34

(6.55)

Taking as an example H0 " 5000 m (high seas) and v0 " 10 cm/s, this yields at the shore (H " 10m) v 10 m/s.

Chapter 5 Sound Waves

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