About the Speed of Sound in Solids Calculator
When a solid is struck or vibrated, its atoms are compressed and stretched locally, and that disturbance propagates outward as a longitudinal (compression) wave — this is what we hear as sound. How fast that wave travels depends on how stiff the material is and how much mass it has to move: stiffer, lighter solids carry sound faster than soft, dense ones. For a slender rod or bar, the longitudinal wave speed is given by v = √(E / ρ), where E is the material's Young's modulus (a measure of stiffness) and ρ is its density.
Deriving the formula
Applying Newton's second law and Hooke's law to a small element of an elastic rod produces a one-dimensional wave equation whose propagation speed is √(E/ρ). This "rod" or "bar" formula is the standard approximation used for thin, slender solids where the material is free to expand sideways as it compresses lengthwise. For a large, unbounded (bulk) solid, sideways expansion is restricted by the surrounding material, and the exact longitudinal wave speed becomes v = √((K + 4G/3) / ρ), where K is the bulk modulus and G is the shear modulus. For most common engineering materials, the rod and bulk values differ by roughly 10-20%.
Using the wavelength and travel-time results
Once the wave speed is known, the wavelength of a sound wave at a given frequency follows from wavelength = v / f — useful for choosing transducer frequencies in ultrasonic testing (typically 0.5-20 MHz for flaw detection) or for predicting how a vibration mode will fit inside a part. Travel time (time = path length / v) is the basis of ultrasonic thickness gauging and time-of-flight defect location: a technician measures how long a pulse takes to travel through a material and multiplies by the known speed to find a thickness or a flaw's depth.