Integrate[  u Sin[ omega ( t - Sqrt[y^2 + u^2]/c )]/Sqrt[ y^2 + u^2], u]

Integrate[  u Sin[ omega ( t - Sqrt[y^2 + u^2]/c )]/Sqrt[ y^2 + u^2], {u, 0, Sqrt[c^2 t^2 - y^2]}]

Simplify[ Cos[t]Cos[y] + Sin[t]Sin[y]]

TrigReduce[ 1- Cos[t]Cos[y] + Sin[t]Sin[y]]

TrigReduce[Sin[x]^2]

Integrate[ Sin[omega t]^2, {t, 0, 2 Pi/omega}]

Plot3D[ Sin[t - Sqrt[y^2]], {t, 0, 10}, {y, -10, 10}, AxesLabel -> {t, y, sin(t - Abs[y])}]

Plot[ {Sin[x], Sin[x-1], Sin[x-2]}, {x, 0, 3 Pi}]