Progressive and regressive wave combination to get the most general form of one dimensional standing wave

Question

Te most general form of a (one dimension) standing wave is $$(A \mathrm{cos}(\omega t)+B \mathrm{sin}(\omega t))(C \mathrm{cos}(k x)+D \mathrm{sin}(k x))=G\mathrm{cos}(\omega t+\phi_1)\mathrm{cos}(kz+\phi_2)\tag{1}$$

Which can be written with exponential notation as

$$G\mathrm{cos}(kz+\phi_2)e^{i(\omega t +\phi_1)}\tag{2}$$

As any standing wave $(2)$ should be the superposition of a progressive and a regressive wave. Nevertheless on textbooks it is usually shown how to get a standing wave like

$$F\mathrm{sin}(\omega t-k x)+F \mathrm{sin}(\omega t+k x)=2F\mathrm{sin}(\omega t)\mathrm{cos}(kz)\tag{3}$$ Or in exponential notation $$F e^{i(\omega t-k x)}+F e^{i(\omega t+k x)}=2F\mathrm{cos}(kz)e^{i\omega t }\tag{4}$$

Which is quite particular, as here $\phi_1=\phi_2=0$.

So what are the progressive and regressive waves (in exponential form) to combine as in $(4)$ but to get an expression as the one in $(2)$ (i.e. where $\phi_1$ and $\phi_2$ are not necessarily zero)?

Answer 1

First note that all we have to do to get $\text{cos} (\omega t + \phi_1)\ \text{cos} (kx+\phi_2)$ from $\text{cos}\ \omega t\ \text{cos}\ kx$ is to measure x from a different origin and t from a different zero of time.

But if we superimpose progressive waves with different phase constants we get $$\text{cos} (\omega t -kx+ \phi_1)\ + \text{cos} (\omega t+kx+\phi_2)= 2\text{cos}\ (\omega t+\frac{\phi_{2}+\phi_{1}}{2})\ \text{cos}\ (kx+\frac{\phi_{2}-\phi_{1}}{2}).$$ So there are your phase constants: $(\frac{\phi_{2}+\phi_{1}}{2})$ and $(\frac{\phi_{2}-\phi_{1}}{2})$.