For a long time, it has been known that the power spectrum of Barkhausen noise had a power-law decay at high frequencies. Up to now, the theoretical predictions for this decay have been incorrect, or have only applied to a small set of models. In this paper, we describe a careful derivation of the power spectrum exponent in avalanche models, and in particular, in variations of the zero-temperature random-field Ising model. We find that the naive exponent, $\left(3\mathrm{}-\tau \right)/\sigma \nu z,$ which has been derived in several other papers, is in general incorrect for small $\tau ,$ when large avalanches are common. $(\tau$ is the exponent describing the distribution of avalanche sizes, and $\sigma \nu z$ is the exponent describing the relationship between avalanche size and avalanche duration.) We find that for a large class of avalanche models, including several models of Barkhausen noise, the correct exponent for $\tau <2\mathrm{}$ is $1/\sigma \nu z.\mathrm{}$ We explicitly derive the mean-field exponent of $2.$ In the process, we calculate the average avalanche shape for avalanches of fixed duration and scaling forms for a number of physical properties.

• Received 15 December 1999

DOI:https://doi.org/10.1103/PhysRevB.62.11699