{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture21

# lecture21 - 2.20 Marine Hydrodynamics Spring 2005 Lecture...

This preview shows pages 1–4. Sign up to view the full content.

Lecture 21 - Marine Hydrodynamics Lecture 21 6.4 Superposition of Linear Plane Progressive Waves 1. Oblique Plane Waves v k k z z k x V p k v = ( z ) x θ x k , k (Looking up the y-axis from below the surface) Consider wave propagation at an angle θ to the x-axis · x k η = A cos( kx cos θ + kz sin θ ωt ) = A cos ( k x x + k z z ωt ) gA cosh k ( y + h ) φ = sin ( kx cos θ + kz sin θ ωt ) ω cosh kh ω = gk tanh kh ; k x = k cos θ, k z = k sin θ, k = k + k x z 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2. Standing Waves + Same A, k, ω , no phase shift η = A cos ( kx ωt ) + A cos ( kx ωt ) = 2 A cos kx cos ωt 2 gA cosh k ( y + h ) φ = cos kx sin ωt ω cosh kh 90 o at all times L , 3 , T T t = 5 3 T T T y x 2A t = , , L 2 2 t = 0, T, 2T, … node amplitude antinode 4 4 4 ∂η ∂φ = · · · sin kx = 0 at x = 0 , = ∂x ∂x k 2 Therefore, ∂φ ∂x = 0. To obtain a standing wave, it is necessary to have perfect x reﬂection at the wall at x = 0. A R Define the reﬂection coeﬃcient as R ( 1). A I y A I = A R x A R R = = 1 A I 2
3. Oblique Standing Waves η I = A cos ( kx cos θ + kz sin θ ωt ) η R = A cos ( kx cos ( π θ ) + kz sin ( π θ ) ωt ) z θ θ I η R η R θ I θ x θ R = π θ I Note: same A, R = 1. k x x k z z ωt η T = η I + η R = 2 A cos ( kx cos θ ) cos ( kz sin θ ωt ) standing wave in x propagating wave in z and 2 π 2 π ω λ x = ; k cos θ V P x = 0; λ z = ; k sin θ V P z = k sin θ Check: ∂φ ∂η x x ∼ · · · sin ( kx cos θ ) = 0 on x

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 11

lecture21 - 2.20 Marine Hydrodynamics Spring 2005 Lecture...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online