= Solution
Use the jump convention $[f]=f_1-f_2$, consistent with region 1 occupying $x>Vt$. The one-dimensional <Lighthill stress tensor> is $T=\rho u^2+p-c_0^2\rho$, so $T=T_2+[T]H(x-Vt)$. Both bulk stresses are constant. The <distributional jump formula for a Heaviside product> therefore gives
$$
\boxed{Q=[T]=\rho_1u_1^2-\rho_2u_2^2+p_1-p_2-c_0^2(\rho_1-\rho_2).}
$$
The <Rankine-Hugoniot condition> for <conservation of mass> is $[\rho u]=V[\rho]$, while <conservation of momentum> gives $[\rho u^2+p]=V[\rho u]$. Hence a physical moving <shock wave> also satisfies
$$
\boxed{Q=(V^2-c_0^2)(\rho_1-\rho_2).}
$$
This sign can be checked directly. Subtracting a constant reference density does not affect derivatives, and $\rho'=\text{constant}+[\rho]H(x-Vt)$. Thus $\rho'_{tt}=V^2[\rho]\delta'(x-Vt)$ and $\rho'_{xx}=[\rho]\delta'(x-Vt)$, exactly reproducing the coefficient above.
Back to article page
