Burgers’ equation. On the other hand, if $P(x,t)>0$, then there are interference effects among the terms in the sums and these lead to rapid oscillations with the effect that $w_\epsilon(x,t)$ no longer converges in the pointwise sense as $\epsilon\downarrow 0$. However, it does converge in the weak topology. The weak limit may be computed by multiphase averaging, which we illustrate in the case $P(x,t)=1$. The procedure is to average the leading term in $w_\epsilon(x,t)$ over an interval in $x$ centered at the point of interest of radius, say, $\epsilon^{p}$ for some $p\in (0,1)$, and then pass to the limit $\epsilon\downarrow 0$. This produces the desired local average over rapid oscillations of wavelength or period proportional to $\epsilon$. Under an ergodic hypothesis that is valid on a set of full measure in the $(x,t)$-plane, this procedure is equivalent to holding $u^\mathrm{B}_n$ and $M_n$ fixed and averaging (with uniform measure) over the torus of relative angles $\phi_1:=\theta_1-\theta_0$ and $\phi_2:=\theta_2-\theta_0$. The double integrals can be evaluated explicitly, with the result that $$\mathop{\mathrm{w}_x\mathrm{-lim}}_{\epsilon\downarrow 0}w_\epsilon(x,t)=
\sum_{n=0}^2 c_n(x,t)u_n^\mathrm{B}(x,t),$$ where $c_n(x,t)$, $n=0,1,2$, are nonnegative coefficients with the property that $c_0(x,t)+c_1(x,t)+c_2(x,t)=1$. Specifically, the coefficients only depend on $x$ and $t$ through $M_0(x,t)$, $M_1(x,t)$, and $M_2(x,t)$. If any of 