I have been applying to PhD programs and asked multiple professors over a month in advance if they would write me letters of recommendation. For two of the programs, one out of the three professors never sent in their letter. One program is already almost three weeks late and the other is a few days late. Note that they did send in letters for other schools, so this has left me confused. I feel pretty disrespected and hurt as I spent quite a bit of time on these applications as well as over 200 dollars. The professor has not returned any of my emails. Note that I cannot or could not have seen them in person as I was out of the country.
With this said, what would the programs do? Will they just throw out my application? I've also resolved to email the programs asking if I could have a different professor write a recommendation letter and send it in. If the programs agreed to this, how should I ask the new professor? They're the head of the department where the other professor who hasn't sent in the rec letter for the two schools works in. I don't want my email to be read as "you weren't my first choice for these programs but I kind of need your help". Would saying something along the lines of "Dear Dr. Professor, One of my letter writers for schools A and B never sent in their letters. After emailing A and B, I am allowed to have another letter sent in and am wondering if you would be able to write one for A and B?". Note that this professor did write letters for other PhD programs that I applied to, so I think it wouldn't be too much of a hassle.