Fourier Transform Solution for an Incident Wave Packet - Revision history

Meylan: /* Solution in the Time Domain using the Frequency Domain Solution */

2010-04-29T22:32:56Z

Solution in the Time Domain using the Frequency Domain Solution

← Older revision		Revision as of 22:32, 29 April 2010
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	<center>		<center>
	<math>		<math>
	\Phi(x,t) = \mathrm{Re} \int_0^{\infty} \bar{f}(\omega) \left\{\phi(x,\omega)e^{\mathrm{i} \omega t}\right\}		\Phi(x,t) = \mathrm{Re} \left\{\int_0^{\infty} \bar{f}(\omega) \left\{\phi(x,\omega)e^{-\mathrm{i} \omega t}\right\}
	\mathrm{d} \omega		\mathrm{d} \omega \right\}
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	\Phi(~~\mathbf{~~x},t) = \mathrm{Re} \int_0^{\infty} \hat{f}(k) \left\{\phi(\mathbf{x},\omega(k))e^{\mathrm{i} \omega(k) t}\right\}		\Phi(x,t) = \mathrm{Re} \left\{\int_0^{\infty} \hat{f}(k) \left\{\phi(\mathbf{x},\omega(k))e^{-\mathrm{i} \omega(k) t}\right\}
	\mathrm{d} k		\mathrm{d} k \right\}
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	f(x) = \mathrm{Re} \int_0^{\infty} \hat{f}(k) e^{-\mathrm{i} k x}		f(x) = \mathrm{Re} \left\{\int_0^{\infty} \hat{f}(k) e^{\mathrm{i} k x} \right\}
	\mathrm{d} k		\mathrm{d} k
	</math>		</math>
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	<center>		<center>
	<math>		<math>
	\hat{f}(k) = \frac{1}{\pi} \int_{-\infty}^{\infty} f(x) e^{\mathrm{i} k x}		\hat{f}(k) = \frac{1}{\pi} \int_{-\infty}^{\infty} f(x) e^{-\mathrm{i} k x}
	\mathrm{d} x		\mathrm{d} x
	</math>		</math>

Meylan at 07:00, 11 April 2010

2010-04-11T07:00:11Z

← Older revision		Revision as of 07:00, 11 April 2010
Line 1:		Line 1:
	{complete pages}		{{complete pages}}

	== Introduction ==		== Introduction ==

Meylan at 06:59, 11 April 2010

2010-04-11T06:59:58Z

← Older revision		Revision as of 06:59, 11 April 2010
Line 1:		Line 1:
			{complete pages}

	== Introduction ==		== Introduction ==

Meylan: /* Solution in the Time Domain using the Frequency Domain Solution */

2010-04-07T02:44:09Z

Solution in the Time Domain using the Frequency Domain Solution

← Older revision		Revision as of 02:44, 7 April 2010
Line 37:		Line 37:
	<center>		<center>
	<math>		<math>
	\Phi(\mathbf{x},t) = \mathrm{Re} \int_0^{\infty} \hat{f}(k) \left\{\phi(\mathbf{x},\omega(k))e^{\mathrm{i} \omega)k) t}\right\}		\Phi(\mathbf{x},t) = \mathrm{Re} \int_0^{\infty} \hat{f}(k) \left\{\phi(\mathbf{x},\omega(k))e^{\mathrm{i} \omega(k) t}\right\}
	\mathrm{d} k		\mathrm{d} k
	</math>		</math>

Meylan: /* Solution in the Time Domain using the Frequency Domain Solution */

2010-04-07T02:43:25Z

Solution in the Time Domain using the Frequency Domain Solution

← Older revision		Revision as of 02:43, 7 April 2010
Line 56:		Line 56:
	<center>		<center>
	<math>		<math>
	\hat{f}(k) = \frac{1}{\pi} \int_{-infty}^{\infty} f(x) e^{\mathrm{i} k x}		\hat{f}(k) = \frac{1}{\pi} \int_{-\infty}^{\infty} f(x) e^{\mathrm{i} k x}
	\mathrm{d} x		\mathrm{d} x
	</math>		</math>
	</center>		</center>



	[[Category:Time-Dependent Linear Water Waves]]		[[Category:Time-Dependent Linear Water Waves]]

Meylan: /* Solution in the Time Domain using the Frequency Domain Solution */

2010-04-07T02:43:09Z

Solution in the Time Domain using the Frequency Domain Solution

← Older revision		Revision as of 02:43, 7 April 2010
Line 47:		Line 47:
	<center>		<center>
	<math>		<math>
	f(x) = \mathrm{Re} \int_0^{\infty} \hat{f}(k) e^{-\mathrm{i} k x~~}\right\~~}		f(x) = \mathrm{Re} \int_0^{\infty} \hat{f}(k) e^{-\mathrm{i} k x}
	\mathrm{d} k		\mathrm{d} k
	</math>		</math>
Line 56:		Line 56:
	<center>		<center>
	<math>		<math>
	\hat{f}(k) = ~~\mathrm{Re}~~ \frac{1}{2\pi} \~~int_0~~^{\infty} f(x) e^{\mathrm{i} k x~~}\right\~~}		\hat{f}(k) = \frac{1}{\pi} \int_{-infty}^{\infty} f(x) e^{\mathrm{i} k x}
	\mathrm{d} x		\mathrm{d} x
	</math>		</math>

Meylan: /* Solution in the Time Domain using the Frequency Domain Solution */

2010-04-07T02:38:09Z

Solution in the Time Domain using the Frequency Domain Solution

← Older revision		Revision as of 02:38, 7 April 2010
Line 21:		Line 21:

	Due to linearity we can write the solution in the time domain as a superposition of the frequency		Due to linearity we can write the solution in the time domain as a superposition of the frequency
	domain solutions		domain solutions. We restrict the potential to the free surface (remembering that the
			surface displacement is proportional to the potential).
	<center>		<center>
	<math>		<math>
	\Phi(~~\mathbf{~~x},t) = \mathrm{Re} \int_0^{\infty} \bar{f}(\omega) \left\{\phi(~~\mathbf{~~x},\omega)e^{\mathrm{i} \omega t}\right\}		\Phi(x,t) = \mathrm{Re} \int_0^{\infty} \bar{f}(\omega) \left\{\phi(x,\omega)e^{\mathrm{i} \omega t}\right\}
	\mathrm{d} \omega		\mathrm{d} \omega
	</math>		</math>

Meylan at 02:37, 7 April 2010

2010-04-07T02:37:04Z

@@ Line 22: / Line 22: @@
 Due to linearity we can write the solution in the time domain as a superposition of the frequency
 domain solutions
 [[Category:Time-Dependent Linear Water Waves]]

Meylan: Created page with '== Introduction == We present here a very general and simple method to solve for an incident wave packet in the time domian using the freqency domain solutions for an incident w…'

2010-04-07T02:29:51Z

Created page with '== Introduction == We present here a very general and simple method to solve for an incident wave packet in the time domian using the freqency domain solutions for an incident w…'

New page

== Introduction ==

We present here a very general and simple method to solve for an incident wave packet in the time domian
using the freqency domain solutions for an incident wave. The theory is present here in two dimensions
but can be extended to three dimensions (for plane incident waves) straightforwardly.

== Frequency Domain Solution ==

{{standard linear problem notation}}

{{standard linear wave scattering equations}}

The simplest case is for a fixed body
where the operator is <math>L=0</math> but more complicated conditions are possible.

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution in the Time Domain using the Frequency Domain Solution ==

Due to linearity we can write the solution in the time domain as a superposition of the frequency
domain solutions

[[Category:Time-Dependent Linear Water Waves]]