Radiation Stress Concerned with the force (or momentum flux) exerted on the right hand side of a plane by water on the left hand side of the plane.

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1 upplement on Radiation tress and Wave etup/et down Radiation tress oncerned wit te force (or momentum flu) eerted on te rit and side of a plane water on te left and side of te plane. plane z "Radiation stress" is te component of tis force due to te waves onl (i.e. minus te drostatic pressure component ut includin te dnamic pressure) or te force eerted on te plane te flu of momentum due to waves. Note: witout eternal forces (viscous dampin, ottom sear stress), te sum of te radiation stress and te drostatic pressure force is constant as te waves o from deep to sallow water is te momentum flu (radiation stress) in te direction of te waves is te momentum flu transverse to te direction of te waves is te flu in te direction of te component of momentum; if waves are perpendicular to te sore, Basic Equations: E n E n [ ( cos θ ) ] [ ( sin θ ) ] Ensin θ were k n, recall c nc sin k if θ (waves perpendicular to te eac): E n [ ] [ ] E n Radiation tress is related to Wave et-up trou te stress alance: ( ) ρ ρ ( )

2 Derivation Force (Momentum Flu) trou a vertical water plane: in direction of waves ( ρu) udz p() z dz mean momentum flu mean pressure force Mc ρ transverse to waves p() z dz ρ, pressure onl Mean Momentum Flu ( ρu ) udz ( ρu) udz ( ρu) n recall: cos ( k t) udz if n odd if n even velocit not defined at z >, terefore use a Talor epansion around z u u (,z, t) u(,) z z... z z were H cos( k t), ( z ) cos k u Hck cos sin k ( k t) ( ρu) udz ρu (,) dz ρu(,) zdz... ρu ρ u ( ρu) udz ( ρu ) (,) ρ u(,) ( 3) z... ρu z > O 3 (,) Acos ( k t) to second order Hck cos udz ρ sin c k 4 ( k t) ( k) cos (,) k ( z ) dz ( z ) cos k( z ) / z sin k sin k k

3 c k sin k cos k sin k k c k cos k sin k k sin k c k sin k k tan k from dispersion relationsip: c tan k c tan k k k tan k sin k tan k k tan k ( ρ ) udz u k sin k cos k k sin k ( ρu ) udz ( ρu) udz ( ρu) udz En nc Mc E c to second order Mean Pressure Force cos k () ( z ) p z ρ ρz, cos k p(z) ρ - z, z ( ) - z p p p () z dz ( p ρz) dz ρ( z) D dz ρ pd dz cos k cos k ρ zdz ρ ρ ( z ) dz ( ) ρ, ( z) dz ρ( ) ρ( ) ρ () z dz ρ ρ ρ O( ) () z dz ρ to second order

4 Total Force (Momentum Flu) trou a vertical water plane: n te wave direction ( ρu) udz p() z dz Mc ρ Force due to drostatic pressure under a wave: F ρ( ) were is te set-up define F ( ) Mc ρ, ρ En ρ En ρ ρ ρ ρ ρ H 6 k sin k H k En ρ ρ 6 sin k k En E ρ, sin k k k n n sin k sin k ( n ) ρ E( n ) O( ) En E ( n E ) to second order Transverse to te wave direction ρ F ρ ρ( ) H k 6 sin k E n ρ E n ρ ρ ( ) ( ) O( ) E ρ k sin k ρ ρ ( n E ) to second order

5 Wave at an anle (θ) to te sore θ Waves ( z ) cos k ' u cosθ Hck cos cos sin k cos k( z ) ' u sin θ Hck cos k t sin sin k u v ( k t) θ ( ) θ ( ρu' ) u'dz p() z dz cos θ ( ρu) udz p() z En cos θ ρ dz En cos θ ρ ρ ( ) En cos θ E( n ) ρ E n( cos θ ) E[ n( cos θ ) ] [ ] O( ) ( ρv' ) v'dz p() z dz sin θ ( ρu) udz p() z En sin θ ρ dz Ensin θ ρ ρ ( ) Ensin θ E( n ) ρ E[ n( sin θ ) ] O( ) [ ( sin θ ) ] E n Now ave component in "" direction wic involves te momentum term onl ( ρu' ) v'dz cosθsin θ ( ρu) ( ρu ) sin θ udz En sin θ θ Ensin udz

6 Mean Mass Flu (M) mean mass flu: M Hc udz cos sin k ρ velocit not defined at z > Talor epansion around z ( k t) Hcos, u ( k t) cos k( z ) dz ( z ) cos k u Hck cos( k t), c tan k sin k u (,z, t) u(,) z z... M ρu M ρ E M c ρu z z (,) dz ρ zdz O() 3 (,) ρ udz ρu ρ z z u 3 (,) z Acos ( k t) ck cos k ck cos ( k t) 4 sin k 4 ck k c c tan k

7 Mean Water Level: Wave et-up and et-down anes in momentum as wave approac sore due to soalin and wave reakin result in a force imalance wic is offset a variation in mean water level know as set-up and set-down ( ). MWL et-down Break point et-up Derivation: φ φ t Bernoulli at te free surface: ( ) ( ) z φ Epand usin Talor series ( st order) aout z and time averain: linear waves: H cos k φ φ φ φ t ( φ ) ( φ ) φ t ( t) cos( kz k) sin( k t) cos( k) k cos( kz k) k cos( k t) H cos( k t) cos( sin( kz k) sin( k t) Hsin( k t) sin( k) sin( kz k) cos( k t) H cos( k t) sin( k) H H H H evaluatin at z & time averain φ φ φ t Hk cos ( k t) H cos ( k t) 4 k [ Hsin( k t) ] H sin ( k t) H H cos 4 k H ( k t) H cos( k t) H cos ( k t) H

8 k H 6 H H k 6 H 6 k H k k 6 k k tan cos sin cos ( ) ( k) tan k ( k) sin( k) ( k) sin ( k) ( k) cos( k) sin( k) sin cos ( k) ( k) H k sin k H k sin k H 6 k sin k deep water: H k 6 sin set-down up to reak point: H k 6 sin k set-up after reak point: 3κ 3κ ( ) were and are values at te reak point, κ H.7 Note: set-up and set-down are equal at te reak point, so H 6 k sin k

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