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Why is the wave formula different in 3-D?

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Why is the wave formula different in 3-D?

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Eh? Because it dependants from four values x,y,z,t and the one dimensional only on two value x,y.

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opps~ i think i didn't make myself clear...

what i meant was why in 3-D does kx become k*r?

what i meant was why in 3-D does kx become k*r?

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Galileo

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isn't r the radius? what does that have to do with the wave vectors, x, y, z?

isn't r the radius? what does that have to do with the wave vectors, x, y, z?

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jtbell

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According to the three-dimensional Pythagorean formula:

[tex]r^2 = x^2 + y^2 + z^2[/itex]

Something that might be causing confusion here is that the formula for a three-dimensional wave depends on the "shape" of the wave. For a plane wave (whose maxima form a series of planes marching through space),

[tex]\psi(x, y, z, t) = A \cos (\vec k \cdot \vec r - \omega t) = A \cos (k_x x + k_y y + k_z z - \omega t)[/tex]

where the [itex]\vec k[/itex] vector and its components are constant.

For a spherical wave (whose maxima form a series of concentric spheres spreading out from a central point, let's say the origin),

[tex]\psi(x, y, z, t) = A \cos (kr - \omega t) = A \cos (k \sqrt{x^2 + y^2 + z^2} - \omega t)[/tex]

At each point in a spherical wave the [itex]\vec k[/itex] vector points radially outward from the origin, so the direction is different everywhere but the magnitude [itex]k = \sqrt {k_x^2 + k_y^2 + k_z^2}[/itex] is constant.

[tex]r^2 = x^2 + y^2 + z^2[/itex]

Something that might be causing confusion here is that the formula for a three-dimensional wave depends on the "shape" of the wave. For a plane wave (whose maxima form a series of planes marching through space),

[tex]\psi(x, y, z, t) = A \cos (\vec k \cdot \vec r - \omega t) = A \cos (k_x x + k_y y + k_z z - \omega t)[/tex]

where the [itex]\vec k[/itex] vector and its components are constant.

For a spherical wave (whose maxima form a series of concentric spheres spreading out from a central point, let's say the origin),

[tex]\psi(x, y, z, t) = A \cos (kr - \omega t) = A \cos (k \sqrt{x^2 + y^2 + z^2} - \omega t)[/tex]

At each point in a spherical wave the [itex]\vec k[/itex] vector points radially outward from the origin, so the direction is different everywhere but the magnitude [itex]k = \sqrt {k_x^2 + k_y^2 + k_z^2}[/itex] is constant.

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how many different kinds of different shapes are there?

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selfAdjoint

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Infinitely many. As many as there are combinations of simple oscillators of different frequency and amplitude.asdf1 said:

how many different kinds of different shapes are there?

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thanks! :)

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