Wanting to understand the linearity of wave equations

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dear yall

with tranditional wave equation on the gre book it says by the linearity in function f which represents wave. it leads to the principle of superposition.

I get an intuition about with a standing wave with cos(x)cos(t) you can break it down to pair of left and right moving waves.

i understand if you sum up the wave is produced from the sum of all subwaves. but how can you get the linearity and such superposition property from simply the wave equation:
grad*grad*f = # d^2f/dt^2

thank you very much
 

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dear yall

with tranditional wave equation on the gre book it says by the linearity in function f which represents wave. it leads to the principle of superposition.

I get an intuition about with a standing wave with cos(x)cos(t) you can break it down to pair of left and right moving waves.

i understand if you sum up the wave is produced from the sum of all subwaves. but how can you get the linearity and such superposition property from simply the wave equation:
grad*grad*f = # d^2f/dt^2

thank you very much
The linearity comes simply from the fact that if ##f_1## and ##f_2## are solutions, then so is ##f_1 + f_2##.

In addition, if ##f## is a solution and ##\alpha## is a number, then ##\alpha f## is also a solution.
 

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