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At the first two specified times, the displacement profiles areThe second curve is the first shifted right by $\pi /(2k)=\lambda /4$. More generally a point of fixed wave phase satisfies $\omega t-kx=\text{constant}$, so it moves along $x=(\omega /k)t+\text{constant}$. Hence the wave travels in the positive $x$ direction withThis phase velocity is distinct from the local transverse velocity $\partial y/\partial t$ of a string element.

For the force derivation take a short segment between $x$ and $x+\Delta x$. Let the tangent angles to the positive $x$ axis be ${\theta }_L$ and ${\theta }_R$. The left tension pulls along the negative tangent and the right tension along the positive tangent. With small slopes and approximately constant tension magnitude $T$, the horizontal components cancel to leading order andThe segment mass is $\rho \Delta x$, where $\rho$ is mass per unit length, not volume density. Newton's second law therefore givesThe approximation neglects bending stiffness, damping and changes in tension caused by finite-amplitude stretching.

Converting the supplied density gives $\rho =0.020\mathrm{kg}{\mathrm{m}}^{-1}$, soFor the pulse, define $g(x)=A{e}^{-a{x}^2/2}$ with $a>0$. The given initial velocity is $-c{g}^{\prime }(x)$, exactly the velocity associated with a right-moving displacement $g(x-ct)$. ThereforeIt moves right without changing shape in this nondispersive model. With the same initial displacement but zero initial transverse velocity, D'Alembert's formula instead gives the Gaussian pulse splitting from zero initial velocityThe two half-amplitude pulses travel in opposite directions. Their initial velocities cancel, while their initial displacements add to the prescribed Gaussian. The two initial-value problems have different kinetic energies; splitting into half-height pulses does not contradict energy conservation. These forms assume a string long enough that reflections from its ends have not returned.