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, the medium must be compressible.

An incompressible fluid behaves like a rigid body. The body moves without deformation and oscillations 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 solid in 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 “acoustic waves” or “sound waves”.

5.2 Sound speed

We consider a front that separates two regions with different pressure and density. The front moves 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 + dp ρ

d x ρ=ρ + ρ

0 0 0 0

0

u =0

0

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

x

0

stationary front

u= c + du u = 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

cρ0A

c du

ρ0 dρ A cρ0A ρ0A du c A dρ or

du c

ρ0

dρ (5.1)

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

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

p₀A

p₀ d p A

net force on V

Aρ0c

c du Aρ0c c

change of x-momentum or

d p ρ0c du (5.2)

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

c² d p dρ

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

ρ .

5.3 Wave equation for sound waves

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

ρ ∂v

∂t

grad p (5.3)

∂ρ

∂t

div

vρ (5.4)

74 CHAPTER 5. SOUND WAVES

5.3.1 Compression waves

We use the decomposition

ρv ρv ₁ ρv ₂ with

curlρv ₁ 0 ρv ₁ gradΦ and

divρv ₂ 0 ρv ₂ curlA

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

grad ˙Φ curl ˙

A grad p (5.5)

and

curl ˙

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˙ div gradΦ div curl

0

A ∆Φ (5.7)

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

Φ¨ p˙ d p

dρ ρ˙ d p dρ∆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 d p dρ. From (5.6) the pressure waves can be computed.

5.3.2 State equation

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

p ρR 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 (or isentrop). 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^" ρ^γ

whereγ c_p c_V is the adiabatic exponent and c_p, c_V is the specific heat under constant pressure and volume, respectively. For a mono-atomic perfect gas one has γ 5 3, for a di-atomic gas γ 7 5.

Thus d p

dρ

γ^" const^" ρ^{γ# 1}

Using (5.8) one determines the constant to RTρ^{1# γ} and finally finds

c^%$ γR T ^&

a value, which is in excellent agreement with the experiment.

76 CHAPTER 5. SOUND WAVES

Chapter 6

Surface Waves

6.1 Preliminaries

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

P₀: external pressure ρ: density of fluid

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x ρ y

z ( )

p

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^.v⁰ 0 ^/ Potential flow

* incompressible fluid, div^.v⁰ 0

77

78 CHAPTER 6. SURFACE WAVES

1 small surface deflection, ² h³ h₀

2

h₀ 4

1

6.2 Gravity waves

6.2.1 equations for flow

Euler equations ⁵ρ⁶ const)

˙

7

v⁸⁹⁵

7

v^: ∇^; v^{7 63} 1

ρgrad⁵ P⁸ U^; with (Potential flow)

7

v⁶ ∇Φ and the formula

5 7

v^: ∇^; v^{7 6} 1 2 <>=?>@∇v²

∇^A∇Φ^B2

3 7

v^CD5∇^C v^{7 ;}

< =? @

E 0

We find

∇ ^FΦ˙ ⁸ 1

2⁵∇Φ^{; 2G 63} ∇ ^F P⁸ U ρ

G (6.1)

or, after integration

Φ˙ ^{6 3} P⁸ U ρ ³

5 ∇Φ^{; 2}

2

∆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 ρgz⁸ U0

6.2.2 Equation for the location of the free surface

Let the surface be located at z⁶ h⁵ x^H y^Ht^;

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^K

h

v (h)z

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

˙h^IL v_xJz^I h^K ∂xh

vx (h)

both together yields (in three dimensions)

˙h^IL N>OP>Qv_xMh

∂xΦ

∂xh^L N>OP>Qv_yMh

∂yΦ

∂yh^R N>OP>Qv_zMh

∂zΦ

(6.3)

or, using the potential:

˙h^IL ∇2Φ^Mh^S ∇2h^R ∂zΦ^Mh (6.4) Now we evaluate eq. (6.2) at the surface z^I h:

Φ˙ ^Mh^IL gJh^L h₀^K

N OP Q

U^Tz^U h^V

L

1

2^J∇Φ^{K 2}_h (6.5)

where we used

U ^I ρgz^R U₀W with U₀^I P₀^L ρgh0

80 CHAPTER 6. SURFACE WAVES

6.2.3 Basic equations and linear solutions

∆Φ ^X 0 (6.6)

Φ˙h ^{X Y} g^Zh^Y h₀[ Y

1

2^Z∇Φ^{[ 2}_h (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 h₀where the fluid is in rest,Φ^X 0 (hydrostatic solution)

Consider small deviations from that state

η^Zx^_t[ X h^Z x^_t[ Y h₀^_ Φ^Zx^_t[ (6.9) eqs. (6.7), (6.8) can be linearized:

Φ˙ ^\z ^{X Y} gη η˙ ^X ∂zΦ^\h ^`

Xba ∂zΦ^{^} 1

g

Φ¨ ^X 0 (6.10)

We assume a solution in form of waves:

Φ^X ξ^Zt[ ] f^Zz[ e^ikx (6.11)

inserting this into (6.6) yields

f^cdceY k²f ^X 0 ^f f^Zz^[bg e^{hjikilkzm h0n}

and with the boundary condition (infinitely deep layer)

Φ^Zz^foY ∞^{[ X} 0 ^f f^Zz[bg e^{ikikzm h0n} (6.12) Substitute (6.11), (6.12) into (6.10) gives

\k^\ζ^{^} 1

g

ζ¨ ^X 0 ^f ζ^Zt[ 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^rg

Thus we have as a solution of the linearized problem

Φ ^p Ae^{skstzu h0v} cos^w kx^x ωt^y (6.14) h ^p h₀^x A k

gsin^wkx^x ωt^y (6.15)

and from there the velocity components

vx ^p ∂xΦ^pz kAe^{skstzu h0v} sin^wkx^x ωt^y (6.16) vz ^p ∂zΦ^p{rk^rAe^{skstzu h0v} cos^wkx^x ωt^y (6.17) This corresponds to traveling waves with the phase velocity

c^p ω k The dispersion relation reads

ω^{p q} kg

Using this, the phase velocity can be expressed as c^p}| g

k ^p

|

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

dx

dt ^p v_x ^pz kAe^{skstzu h0v} sin^wkx^x ωt^y (6.18) dz

dt ^p v_z ^p}rk^rAe^{skstzu h0v} cos^wkx^x ωt^y (6.19) two coupled nonlinear ODE’s which can be solved only numerically.

82 CHAPTER 6. SURFACE WAVES Approximation: ^~dv^~€ c

With the initial condition x₀‚ x^ƒ t‚ 0^„ z₀‚ z^ƒt ‚ 0^„ one can integrate

x^ƒt^{„ ‚} x0^† t

‡

0

vx^ƒx0z0 t^ˆ‰„ dt^{ˆ ‚} x0^† a^ƒcos^ƒkx0^Š ωt^{„ Š} cos kx0^„ (6.20)

z^ƒt^„ ‚ z₀†

t

‡

0

v_z^ƒx₀ z₀ t^ˆ„ dt^ˆ ‚ z₀Š b^ƒsin^ƒkx₀Š ωt^{„ Š} sin kx₀^„ (6.21)

with a ‚ Š

Ak

ωe^{‹k‹lŒz0 h0Ž} (6.22)

b ‚

A^~k^~

ω e^{‹k‹Œz0 h0Ž} (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

~dv_s^~ 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 harmonic waves but can be any other form. It is important that the dimension (extension) in horizontal direction 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 ^‘ ∂²_˜z˜zΦ˜ ^™ δ²∂²_{x ˜˜x}Φ˜ (6.26) Φ˙˜ ^{‘ š} G ^› ˜h^š 1^œš 1

2^›∂x˜Φ˜^{œ 2š} 1

2δ^2› ∂˜zΦ˜^{œ 2} (6.27) δ²˙˜h ^{‘ š} δ²∂x˜Φ˜ ^– ∂x˜˜h^™ ∂˜zΦ˜ (6.28) (from here, we suppress the tildes). The non-dimensional number

G^‘ g^– h₀^– τ²

’ 2

84 CHAPTER 6. SURFACE WAVES is called “gravitation number”.

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

Φ^ž Φ^Ÿ0 ˜¡ δ²Φ^Ÿ2 ˜¡ δ⁴Φ^{Ÿ4 ˜¡£¢¤¢¤¢} (6.29)

this inserted into (6.26) gives:

∂²_zzΦ^Ÿ0 ˜¡ δ^2¥ ∂²_zzΦ^Ÿ2 ¦¡ ∂²_xxΦ^{Ÿ0 ¨§©¡} δ^{4 ¥} ∂³_zzΦ^Ÿ4 ¦¡ ∂²_xxΦ^Ÿ2 ¨§©¡ª¢¤¢¤¢

ž 0 (6.30)

Sinceδcan be arbitrary, terms with the same order ofδmust vanish:

« orderδ⁰

∂²_zzΦ^{Ÿ0  ž} 0 ^¬ Φ^{Ÿ0  ž} f₁­x^®t^{¯ ¡} f₂­ x^®t^¯° z (6.31) with the boundary condition on the ground (z^ž 0)

v_z­ z^ž 0^¯bž ∂Φ

∂z ^±

±±±

z^² 0

ž f₂^ž 0 ^³ f₂^ž 0 (6.32)

and finally the important result

Φ^{Ÿ0  ž} Φ^{Ÿ0  ­} x^®t^¯ (6.33)

« orderδ²:

∂²_zzΦ^{Ÿ2  ž ´} ∂²_xxΦ^{Ÿ0  ž´} Φ^Ÿ0 ¶µµ (6.34)

¬ Φ^{Ÿ2  ­} x^® z^®t^{¯·ž ´} Φ^{Ÿ0 µµ °} z²

2

¡ f₃­ x^®t^¯

¸ ¹º »

² 0 (b. c)

°z^¡ f₄­ x^®t^¯ (6.35)

« orderδ⁴

- in the same way ^¢¤¢¤¢

We write down the result up to the orderδ⁴:

6.3. THE SHALLOW WATER EQUATIONS 85

Φ^¼x^½ z^½t^¾·¿ Φ^{À0Á ¼}x^½ t^¾Ã δ^{2 ÄÆÅ} Φ^{À0Á¶ÇÇÉÈ} z²

2 ^ f₄^¼x^½ t^¾ÆÊ

 δ^{4 Ä}Φ^{À0Á¶ÇÇÇÇeÈ} z⁴

24

Å f^ËdË₄ ^È z²

2 ^ f₆^¼ x^½t¾ÆÊÌÂªÍ¤Í¤Í (6.36)

– we know the z-dependence ofΦexplicitly !!

– ifΦ^{À0Á ¼} x^½t^¾ is known,Φ^¼x^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˙^{À0Á ¿ Å} G^¼h^Å 1^{¾ Å} 1

2 ^Î ∂xΦ^{À0Á¨Ï 2} (6.37)

˙h ^{¿ Å} ∂xΦ^À0ÁÐÈ ∂xh^Å h^È ∂²_xxΦ^À0Á (6.38) or in two horizontal dimensions (x^½ y)

Φ˙ ^{¿ Å} G^¼ h^Å 1^{¾ Å} 1

2^¼∇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 deviationsηfrom the constant depth h0^¿ 1:

86 CHAPTER 6. SURFACE WAVES

η^Ò h^Ó 1

ThenΦis 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 ˙Φyields a wave equation forη

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. THE SHALLOW WATER EQUATIONS 87

6.3.2 Numerical solutions of the nonlinear Shallow Water equations

time evolution in 1D

x t

x t

h

snapshot in 2D

88 CHAPTER 6. SURFACE WAVES

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:

ˆ

n^ØÚÙv^Û nˆ^Ø ∇Φ^Û 0

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

Φ˙ ^{Û Ü} G^Ýh^Ü 1^ÞßÜ 1

2^Ý∇2Φ^{Þ 2} (6.43)

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

linearizing again yields a wave equation, now of the form

¨h^Ü G∇2^ãH^Ý x^äy^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 to small changes of the surface on the length scale of the waves), (6.45) describes waves with space dependent velocity

c_p^Ýx^ä y^ÞæÛ{ç G H^Ýx^ä y^Þ (6.46)

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

λ^Û 2π k ^Û

2π

ωc_p^Û 2π

ω ^è GH ^é H¹² (6.47)

6.3. THE SHALLOW WATER EQUATIONS 89 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 (see textbooks, 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 shall return to this issue in the sect. on Tsunamis.

90 CHAPTER 6. SURFACE WAVES

6.3.4 Generation of waves by a time-dependent ground

z

x h(x,t)

z=1 n ^

f(x,t) H(x,t)

The boundary conditions now have the form:

ˆ

n^ì¤ív^î nˆ^ì ∇Φ^î f˙ This gives another term (underlined)

Φ˙ ^{î ï} G^ð h^ï 1^ñï 1

2 ^ð∇2Φ^{ñ 2} (6.49)

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

IfΦ^î const (fluid in rest) ^ô ˙h^î f˙ ^ô h^ðt^ñõî f^ðt^{ñ ó} const

ö the ground motion is equal to the surface motion

ö no time delay, reason: fluid is assumed to be incompressible

ö A ground motion may generate waves:

linearized wave equation with H ^î 1^ï f : 1

G

Φ¨ ^ï ∇2^÷H∇2Φ^øÃî f˙^îï H˙ (6.51) This is an inhomogeneous wave equation. It can be formally solved using an appropriate Green’s function:

Φ^ðx^ù y^ù t^{ñõîûúüú} dx^ýdy^ýÚú dt^ý D^ð x^ï x^ýþùy^ï y^ýÿùt^ï t^ý‰ñ H˙^ðx^ýÿùy^ýù t^ýñ (6.52)

6.3. THE SHALLOW WATER EQUATIONS 91 Example

consider the following localized ground motion:

H

x y t

1 v₀ tδx δy 0 t t₀

1 v₀ t₀δx δy t₀ t (6.53)

0 < t < t₀ t = t t > t₀

0

ground t = 0

solution: circular waves

t=4

t=6

t=8 φ t=2

r

snapshots at various times t

92 CHAPTER 6. SURFACE WAVES

Numerical solution in two dimensions

We chose

fxyt 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. THE SHALLOW WATER EQUATIONS 93

6.3.5 Tsunamis

The notion “Tsunami” was coined by Japanese fishermen and means “wave in harbor”. The fishermen went out to the sea during the night for fishing. On their return, they found the harbor destroyed by a flood. Since they didn’t notice anything unusual on the open sea, they thought that 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 23 meters

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 are short waves or deep water waves.

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

ground surface

seaquake

Thus, the fluid over the whole depth moves. Although the fluid motion is rather slow compared to 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.

94 CHAPTER 6. SURFACE 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/s²and 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 A h₀ c with A = 50 cm, h0= 4000 m, c 200 m/s one gets

6.3. THE SHALLOW WATER EQUATIONS 95

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 not moving (or only in large scaled streams).

For the frequency, we can estimate

ν^" ω

2π ^"

c^# k 2π ^"

c λ

Takingλ^" 100 km and c^" 200 m/s one hasν 0^$002 Hz, corresponding to∆t 500 s between two 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^" A

H c

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

v v₀ ^"('

H₀ H ⁾

3

4 (6.55)

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