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Physics of Surfing
Waves - part 2David T. Sandwell(http://topex.ucsd.edu/ps)
• Physics of waves
• Characteristics of waves
• Generation of waves by storms
• Wave speed - shallow vs. deep ocean
• Sets - dispersion
• Refraction of waves - Why is Black!s so good?
Exercises: April 15(each problem is covered in class)
1. Derive the expression for the period of a harmonic oscillator with mass mand spring constant k.
2. Derive the expression for the speed of a deep water wave in terms of thewave period T.
3. What are sinh, cosh, and tanh in terms of the exponential function? Whatis tanh(10-6)? What is tanh(10)?
4. One more problem. Consider two waves of equal height but differentperiod (T1=12.5 s and T2=13 s. What is the time between sets? Here is ahint: add two cosine functions h(t) = cos(!1*t) + cos(!2*t) where !1 = 2*"/T1 ,use the trigonometric formula for the sum of two cosines, theninterpret or plot the results.
wave characteristics
• generated by storms at sea
• far from the storm they are sinusoidal
h
xH
A
L
�
h(x, t) = Asin2!xL
"2!tT
#
$ %
&
' (
c =L
T phase velocity
L - wavelength
A - amplitude
H - height
T - period (5 - 18 s)
Airy solution
L - wavelength
g - acc. gravity
d - ocean depth
�
c(d) =gL
2!tanh
2!dL
"
# $
%
& '
(
) *
+
, -
1/ 2
deep water waves shallow water waves
�
d >> L /2
cd =gL
2!
�
d << L /2
cs = gd
What causes “sets”?
44 mph
60 mph
Munk theory of sets
D RSan Diego
Antarctic
storm
Waves arrive in San Diego at the same time t1.
Suppose the waves were generated at the same time to.
�
t1! t
o= 2R /c
1
t1! t
0= 2(R + D) /c
2
c2
= c1
R + D
R
"
# $
%
& '
�
c =gT
2! deep water dispersion
T2 = T1
R + D
R
"
# $
%
& '
Munk theory of sets
D RSan Diego
storm
�
h(t) = Acos2!tT1
"
# $
%
& ' + Bcos
2!tT2
"
# $
%
& ' , suppose B = A
h(t) = 2Acos !t1
T1
+1
T2
"
# $
%
& '
(
)
*
+
,
- cos !t1
T1
.1
T2
"
# $
%
& '
(
)
*
+
,
-
surf = mean period modulated by beat period
interval between sets
�
TB
= T1
1!R
R + D
"
# $
%
& '
!1
R = 7000 km D = 400 km T1
= 17s,
TB
= 5.5min
A long time to wait
between sets!
“sets”
D RSan Diego
Antarctic
storm
A=3 m, B=1m, sets every 5.5 min
OCEANSIDE OFFSHORE, CA - Station:
04501
Water depth(m): 220.00
August 3, 2007
Average time between sets
8.8 min
cm
shallow water waves
�
cs = gd
wave refraction
60 ft
35 mph
24 ft
22 mph15 ft
18 mph4.5 ft
10 mph
ocean depth and breaker height - empirical
Hb - height of breaker
db - depth where wave
breaks
�
db
=1.28Hb
Conclusions
– Ocean waves: force of acceleration is balanced by the forceof gravity.
– Wind speed >= wave speed. 17-s period waves require windspeed of 27 m/s = 60 mph.
– Wave speed: d >> L/2 waves are dispersive; d << L/2 speeddepends on depth.
– Refraction is important when d < 5L.