> [!NOTE] > The [[Wave Equation in 1 Spatial Dimension|1D wave equation]] $\partial_{tt}y=c^2 \partial_{xx}y$ describes evolution of a plucked string is given by where $y(x,t)$ is the vertical coordinate of the point at $x$ over time $t.$ **Derivation**: Consider a flexible string that is stretched to a tension $T$ and has mass density $\rho.$ The waves we are interested in are transverse vibrations that cause a point initially at (x,0) to be displaced to $(x,y(x,t)).$ The velocity vector is $\mathbf{v}=(0,\partial_ty)$ and the tangent vector to the string is $\boldsymbol{\tau}=(1,\partial_xy).$ We assume the vibrations are small in the sense that $|\partial_xy|$ is small. Hence, $\boldsymbol{\tau}$ is approximately a unit vector. Consider an arbitrary section $a\leq x\leq b$ of the string. The force exerted by the string at a given point is $T\boldsymbol{\tau}(x,t).$ Newton's second law says that the change in momentum is the difference between this force at either end: $\frac{d}{dt}\int_a^b\rho\mathbf{v}(x,t)\:\mathrm{d}x=T\boldsymbol{\tau}(b,t)-T\boldsymbol{\tau}(a,t).$ This is an equation of vectors, however the first entry is uninteresting (it says that zero equals one minus one). Keeping only the second entries, we get the scalar equation$\frac{d}{dt}\int_a^b\rho\partial_ty(x,t)\:\mathrm{d}x=T\partial_xy(b,t)-T\partial_xy(a,t).$ Assuming $y$ is sufficiently smooth, we reach$\int_a^b\rho\partial_{tt}y(x,t)\:\mathrm{d}x=\int_a^bT\partial_{xx}y(x,t)\:\mathrm{d}x.$ Since $a$ and $b$ are arbitrary, this leads to the wave equation$\partial_{tt}u=c^2\partial_{xx}u$where $c=\sqrt{T\rho^{-1}}$ is the speed of the waves on the string.