Heat equation: interpretation of Fourier series constants

Question

I am pretty aware of the interpretation of the equation itself: $u_t=\lambda u_{xx}$ along with the boundary conditions $u(0,t)=u(l,t)=0$ and $u(x,0)=T(x)$, with $0<x<l$ (e.g., Gyu's answer Interpretation of the heat equation is an excellent explanation).

My question is something about the Fourier series that describes the initial temperature $T(x)=u(x,0)$. When solving the equation (along with the boundary conditions), one realizes that the function $T(x)$ can be described as a Fourier series:

$$T(x)=\sum_{n=1}^{\infty} A_n \sin(\frac{n\pi x}{l}),$$

with $0<x<l$. The $A_n$ can be completely described by integrating $\frac{2}{L}\int_0^l f(x)\sin(\frac{n\pi x}{l})dx$, which can be used to described the final solution of the PDE, namely,

$$u(x,t)=\frac{2}{l}\sum_{n=1}^{\infty} \bigg(\int_0^l f(x)\sin(\frac{n\pi x}{l})dx\bigg)e^{-\lambda((n^2\pi^2)/l^2)t}\sin(\frac{n\pi x}{l}).$$

Is there any physics interpretation of the $A_n$ in terms of this specific problem? Moreover, is there any physics interpretation of the functions $A_n \sin(\frac{n\pi x}{l})$.

My questions are based on the idea that if we have a function that describes some vibration (like some sound) then its Fourier series can be understood as a decomposition of vibrations that caused the vibrations of $f(x)$ by overlapping (and then I can study the lower/higher frequencies). This idea is used as an application for several problems, so I was wondering if there is something similar to it in the heat equation.

Thanks

Answer 1

In a first place the Fourier series is only a useful mathematical tool to analytically solve or approximate a partial differential equation (PDE). One should be careful to attribute too much physical meaning to these mathematical expressions.

Next you can regard the Fourier series as a trigonometric interpolation, which is an exact approximation of the PDE in the steady state $T(x)$ for $n\rightarrow\infty$. Here each individual summand can be intepreted as mode with a prefactor $A_n$ as weight. For example the first mode $A_0$ is the integral mean.

Note that the polynomial representation with a trigonometric basis is only one of many modal representations. You could also use other arbitrary polynomials. The interpretation of the weights would change each time.

Regards