Equation of a Plane Progressive Wave

What this figure shows. A single bump-shaped pulse is drawn on a y-versus-x axis in panel (a) at time t = 0, with its peak located at the origin O and a point P marked on its leading edge. In panel (b), the identical pulse shape is drawn again, but now shifted bodily to the right by a distance vt, with the same point P now relabelled as having moved to a new position, and an arrow of length vt marking exactly how far the whole pulse (and point P along with it) has advanced. The figure is the direct visual basis for writing the travelling-pulse equation $y(x,t)=f(x-vt)$: it shows that the pulse's shape itself is completely unchanged between the two snapshots, and that the entire pattern has simply translated rightward by the distance vt, so that a coordinate measured as $x'=x-vt$ in …

Worked out. The straight line $y=x-a$ is plotted for several successive values of the constant a (such as a = 0, 1, 2, 3), and the resulting family of parallel lines is examined to see how the line's position changes as a increases. Since $y=x-a$ is simply the line $y=x$ shifted so that it crosses the x-axis at $x=a$ instead of at the origin, increasing a moves the line's x-intercept progressively further to the right while its slope (always 1) stays unchanged, so the whole line visibly shifts rightward as a increases. This purely geometric observation -- that increasing the parameter a in an expression of the form $y=f(x-a)$ simply translates the graph of f rightward by a -- is exactly the mathematical mechanism used one section later to justify why $y=f(x-vt)$, with $a=v …

Worked out. The curve $y=\sin(x-a)$ is sketched for five successive values of a: $a=0,\ \pi/4,\ \pi/2,\ 3\pi/4,\ \pi$, all plotted together on the same x-axis running from 0 to $2\pi$. Each successive curve is seen to be an exact copy of the plain sine curve $y=\sin x$, but shifted progressively further to the right as a increases through each of these five values, exactly mirroring the straight-line shifting behaviour already established in the previous example. Taking a to represent $vt$ for a steadily increasing time t (with $v=\pi/4$, say) confirms explicitly that $y=\sin(x-vt)$ describes a sinusoidal shape travelling steadily in the positive x-direction as time advances, while the complementary form $y=\sin(x+vt)$ would …

Worked out. The expression $y=\sin(x-vt)$ is checked for dimensional consistency, since the sine function can only ever take a dimensionless argument. Here $x$ has dimension of length and $vt$ also has dimension of length (speed times time), so the difference $x-vt$ genuinely does have a consistent dimension -- but that dimension is length, not the required dimensionless quantity, so the expression as written is dimensionally WRONG despite $x$ and $vt$ matching each other. The corrected, dimensionally consistent form multiplies $x$ by the wave number $k$ (dimension inverse-length) and $t$ by the angular frequency $\omega$ (dimension inverse-time), giving $y=\sin(kx-\omega t)$, which can equivalently be written using wavelength and period as $y=\sin\!\big(2\pi x/\lambda - 2\pi t/T\big)$; this is exactly the standard sinusoidal progressive-wave form $y(x,t)=A\sin(kx-\omega t)$ used throughout the rest of the chapter, with its parts labelled: A is the amplitude, $(kx-\omega t)$ is the phase, k …