🎬visualize the wave equation solution for the condition where the string starts with zero initial dis

visualize the wave equation solution for the condition where the string starts with zero initial displacement but is given a velocity profile

The demonstration visualizes the special case of the 1D wave equation where the string has zero initial displacement but is struck with a finite initial velocity profile ( $\partial_t u(x, 0)=V_0(x)$). The key takeaway is the resulting anti-symmetric wave splitting: the initial velocity disturbance immediately resolves into two distinct, permanent pulses-one positive peak traveling to the right ( $+H(x-c t)$ ) and one negative trough (of equal magnitude) traveling to the left ( $-H(x+c t)$ ). This contrasts sharply with the "released from rest" case, which produces two identical symmetric peaks, proving that the nature of the initial energy input (displacement vs. velocity) fundamentally dictates the symmetry of the resulting traveling wave components .

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visualize the wave equation solution for the condition where the string starts with zero initial displacement but is given a velocity profile | Animated Resultsviadean on Notion