Step 1: Definition through Maxwell's idea.
A changing electric field creates a magnetic field and a changing magnetic field creates an electric field; this mutual regeneration lets a combined electric-magnetic disturbance travel forward as an electromagnetic wave. Both fields are in phase, perpendicular to each other, and perpendicular to the travel direction, and in vacuum the wave moves at \(c = 1/\sqrt{\mu_0\varepsilon_0} = 3\times10^8\) m/s.
Step 2: Fix the right-handed triad.
Propagation is along \(X\), so \(\vec{E}\) and \(\vec{B}\) must lie in the \(Y\)-\(Z\) plane. Given \(\vec{E}\) along \(Y\) and \(\vec{B}\) along \(Z\), the set \((\vec{E}, \vec{B}, \text{direction of motion})\) forms a right-handed system since \(\hat{Y}\times\hat{Z}=\hat{X}\).
Step 3: Write the travelling-wave forms.
\(E_y(x,t) = E_0\cos(kx - \omega t)\) and \(B_z(x,t) = B_0\cos(kx - \omega t)\), with \(E_0 = cB_0\), \(k = 2\pi/\lambda\) and \(\omega = ck\). A sine or cosine form is equally valid.
Step 4: Contrast with sound.
Sound is a longitudinal mechanical wave that needs air, water or a solid to travel and moves at roughly \(340\) m/s in air; an EM wave is a transverse non-mechanical wave that travels even through vacuum at \(3\times10^8\) m/s and carries oscillating fields rather than moving matter. Sound cannot be polarised, whereas EM waves can, precisely because they are transverse.
\[\boxed{E_y = E_0\cos(kx-\omega t),\quad B_z = B_0\cos(kx-\omega t),\quad E_0 = cB_0}\]