= Solution
The result depends on the component <amplitudes> and <wave phases>; adding many <frequencies> does not by itself guarantee one localized <wave packet>. For a definite sketch, take many equally spaced <wavenumbers> $k_j=k_0+j\Delta k$ in the same fixed band, with equal, aligned <complex amplitudes> $B/N$. Linearize the <dispersion relation> as $\omega(k_j)\simeq\omega_c+v_g(k_j-k_0)$, where $\omega_c=k_0^2/m$. Then with $\xi=x-v_gt$ the <wave envelope> is
$$
\frac BN\sum_{j=-(N-1)/2}^{(N-1)/2}e^{ij\Delta k\xi}
=\frac BN\frac{\sin(N\Delta k\xi/2)}{\sin(\Delta k\xi/2)}
$$
for odd $N$. This is a normalized <Dirichlet kernel>: <constructive interference> occurs near $\xi=0$, while most other positions undergo partial cancellation. At finite $N$ there are repeated groups separated by $2\pi/\Delta k$.
As $N$ grows with the half-bandwidth $K$ fixed, $\Delta k$ decreases and the repetition spacing grows. The continuous flat-band limit is
$$
\boxed{y(x,t)\simeq Be^{i(k_0x-\omega_ct)}\frac{\sin(K\xi)}{K\xi},\qquad \xi=x-v_gt.}
$$
Thus a <sinc function> envelope has a central group between its first zeros at $\xi=\pm\pi/K$, smaller side lobes and rapidly oscillating crests. At fixed bandwidth the limiting central width is $2\pi/K$; it does not shrink indefinitely just because more components are added. Smoother spectral weights can reduce the side lobes, and random <wave phases> would produce a different shape.
\Image[/past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2021-wave-packet.png]
{title=Equal-phase waves in a fixed band: a two-tone beat train, finite groups, and the continuous sinc envelope}
For the exact quadratic <dispersion relation>,
$$
y(x,t)=\frac{B}{2K}e^{i(k_0x-\omega_ct)}\int_{-K}^K e^{i[q(x-v_gt)-q^2t/m]}\,dq.
$$
The additional quadratic <wave phase> makes the <wave packet> spread and change shape. The translating <sinc function> approximation is valid while $|t|K^2/|m|\ll1$; the preceding figure is the exact $t=0$ comparison, where that approximation introduces no error.
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