Recipe
Evaluate the wave equation's Green's function at a point
Use when: you have wave speed c, a source at (x_0,t_0), and a target point (x,t), and need G(x,t;x_0,t_0)
Goal: the value 1/(2c) or 0, decided from the wedge condition alone, with no delta-function integration by hand
Taught in The Green's function for the wave equation
Quick recipe
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1
Compute c(t-t_0): the distance a wave from the source could have traveled by time t.
Example
With c=2, t_0=0, t=2: the wedge's reach is c(t-t_0)=2(2)=4.
Why: this distance is exactly the radius of the source's own range of influence at time t.
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2
Compute |x-x_0|, the target point's distance from the source, and compare it to the reach from the last step.
Example
At x=3, x_0=0: |3-0|=3<4, inside. At x=5: |5-0|=5\geq4, outside.
- |x-x_0|<c(t-t_0) — the point sits inside the wedge; continue to the next step for the nonzero value.
- |x-x_0|\geq c(t-t_0) — the point sits outside the wedge; G=0 immediately, no further work needed.
Why: the delta function's mass is picked up only when x_0 falls inside the interval of integration that defines G.
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3
When the point is inside, G takes the single fixed value 1/(2c). It doesn't depend on how deep inside the wedge the point sits, or on t itself.
Example
For c=2 and the point from Step 2 that landed inside: G(3,2;0,0)=1/(2\cdot2)=1/4.
Why: the unit impulse's entire mass lands at one instant, so once it's inside the integration window the value is that one constant.
Common mistakes
- Assuming G shrinks as t-t_0 grows, the way the heat kernel does; the wave Green's function stays a constant height inside the wedge for every t>t_0.
- Applying < and \geq inconsistently right at the wedge's exact boundary.
- Forgetting that G=0 identically for t\leq t_0, the same causality requirement every time-dependent Green's function must satisfy.
- Writing 1/c instead of 1/(2c) for the constant value inside the wedge.